Special Values of Zeta-Functions for Proper Regular Arithmetic Surfaces Citation Siebel, Daniel A. (2019) Special Values of Zeta-Functions for Proper Regular Arithmetic Surfaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/YMHN-2T74. https://resolver.caltech.edu/CaltechTHESIS:11142018-032432585 Abstract We explicate Flach's and Morin's special value conjectures in [8] for proper regular arithmetic surfaces π : X → Spec Z and provide explicit formulas for the conjectural vanishing orders and leading Taylor coefficients of the associated arithmetic zeta-functions. In particular, we prove compatibility with the Birch and Swinnerton-Dyer conjecture, which has so far only been known for projective smooth X. Further, we derive a direct sum decomposition of Rπ * Z(n) into motivic degree components. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Special Values, Zeta-Functions, Arithmetic Surface Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Apostol Award for Excellence in Teaching in Mathematics, 2015. Thesis Availability: Public (worldwide access) Research Advisor(s): Flach, Matthias Thesis Committee: Mantovan, Elena (chair) Flach, Matthias Graber, Thomas B. Amir-Khosravi, Zavosh Defense Date: 9 November 2018 Record Number: CaltechTHESIS:11142018-032432585 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11142018-032432585 DOI: 10.7907/YMHN-2T74 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 11273 Collection: CaltechTHESIS Deposited By: Daniel Siebel Deposited On: 10 Dec 2018 22:13 Last Modified: 04 Oct 2019 00:23 Thesis Files Preview PDF - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

Special Values of Zeta-Functions for Proper Regular Arithmetic Surfaces Thesis by Daniel A. Siebel In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2019 Defended November 9, 2018 ii c 2019 Daniel A. Siebel All rights reserved. iii Acknowledgements First and foremost, I would like to thank my advisor, Matthias Flach, for his guidance and support throughout my time as a graduate student. I am very grateful for his help, for his many ideas and countless discussions, and for his patience and kindness which have been essential for me as well as my research. I would also like to give thanks to my parents, my brother, and my grandparents for their endless support and care over the years. Lastly, I thank the California Institute of Technology for going to great lengths to support their students, and its Math Department for their warm and welcoming environment and for the opportunity to pursue mathematical research. iv Abstract We explicate Flach’s and Morin’s special value conjectures in [8] for proper regular arithmetic surfaces π : X Ñ Spec Z and provide explicit formulas for the conjectural vanishing orders and leading Taylor coefficients of the associated arithmetic zeta-functions. In particular, we prove compatibility with the Birch and Swinnerton-Dyer conjecture, which has so far only been known for projective smooth X . Further, we derive a direct sum decomposition of Rπ˚ Zpnq into motivic degree components. v Contents Acknowledgements iii Abstract iv 0 Introduction 1 1 Artin-Verdier duality for arithmetic surfaces 5 1.1 The Artin-Verdier ètale topos and compact support cohomology . . . . . . . . 5 1.2 The Duality Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Lichtenbaum’s product map and Spiess’ complex . . . . . . . . . . . . . . . . 9 1.4 A further construction of the p-torsion product map for arithmetic surfaces a la Sato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Saito’s duality result and duality on the closed part . . . . . . . . . . . . . . . 14 1.6 Duality on the open part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.7 Proof of the global duality statement . . . . . . . . . . . . . . . . . . . . . . . 19 2 Motivic decompositions of cohomology for arithmetic surfaces 22 2.1 L- and ζ-functions associated to arithmetic surfaces . . . . . . . . . . . . . . . 23 2.2 Motivic decompositions of push-forward sheaves in the presence of a section . 25 2.2.1 Verdier Duality and and Cohomological Purity . . . . . . . . . . . . . 25 2.2.2 A motivic decomposition for push-forwards of constant torsion sheaves 27 2.2.3 A motivic decomposition for Rπ˚ Zpnq . . . . . . . . . . . . . . . . . . 29 2.2.4 Motivic decompositions for complex manifolds. . . . . . . . . . . . . . 34 2.2.5 The motivic picture and notation . . . . . . . . . . . . . . . . . . . . . 34 2.3 Deligne Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4 Étale motivic cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4.1 Completed motivic cohomology . . . . . . . . . . . . . . . . . . . . . . 37 2.4.2 The case n “ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4.3 Compact support cohomology and the perfect pairing conjecture . . . 38 2.4.4 Torsion of motivic cohomology . . . . . . . . . . . . . . . . . . . . . . 40 vi 2.4.5 Weil-étale cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5 Betti cohomology and Weil-étale cohomology with compact support . . . . . . 42 2.6 De Rham and derived de Rham cohomology . . . . . . . . . . . . . . . . . . . 43 2.7 Completions of L- and ζ-functions . . . . . . . . . . . . . . . . . . . . . . . . 52 3 The special value conjectures in their Weil-étale formulation 55 3.1 Conjectural vanishing orders and compatibility with BSD . . . . . . . . . . . 56 3.2 Integral Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 The Regulator RpX q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4 The Fundamental Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4.1 The Trivialization Factor for n “ 1 . . . . . . . . . . . . . . . . . . . . 69 3.4.2 Trivialization Factors for n ě 2 . . . . . . . . . . . . . . . . . . . . . . 71 3.4.3 Trivialization Factors for n ď 0 . . . . . . . . . . . . . . . . . . . . . . 73 3.5 The Correction Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.6 The Functional Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.7 Summary of special value results . . . . . . . . . . . . . . . . . . . . . . . . . 84 A Computational Material 86 A.1 The motivic cycle complexes Zpnq . . . . . . . . . . . . . . . . . . . . . . . . . 86 A.2 GR -equivariant cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 A.3 Comparison between motivic and completed motivic cohomology . . . . . . . 94 A.4 Supplementary material for derived de Rham cohomology . . . . . . . . . . . 97 A.5 Overview of computed cohomology groups . . . . . . . . . . . . . . . . . . . . 100 A.5.1 Rank tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 A.5.2 Motivic and Weil-étale motivic cohomology tables . . . . . . . . . . . 102 A.6 Diagram of cohomology groups . . . . . . . . . . . . . . . . . . . . . . . . . . 104 A.7 Overview of integral structures occurring in ∆pX {Z, nq . . . . . . . . . . . . . 105 B Bibliography 106 1 Chapter 0 Introduction Background. The Tamagawa Number Conjecture — first proposed by Bloch and Kato in [4] and then reformulated by Fontaine and Perrin-Riou in [9] — describes the vanishing order and leading Taylor coefficient (up to sign) L˚ pM, nq of the L-function LpM, sq associated to any Chow motive M over a number field at every integer s “ n. It vastly generalizes the Analytic Class Number Formula as well as the Birch and Swinnerton-Dyer Conjecture which could be derived as corollaries for M “ h 0 pSpec F q for any number field F and for M “ h 1 pEqp1q for any elliptic curve E (or, more generally, any smooth projective curve) over F respectively. Meanwhile Flach and Morin gave conjectural descriptions of the special values of arithmetic ζ-functions ζpX , sq associated to any proper regular arithmetic scheme X Ñ Spec Z in [8]. They proved that under certain standard assumptions their conjectures are compatible with the Tamagawa Number Conjecture for projective smooth X , in the sense that they predict the same vanishing orders and leading Taylor coefficients for the L-function associated to the Tate-twisted motive M “ hpXQ qpnq of the generic fiber XQ (cf. [8] Thm. 5.26). Special Value Conjectures for arithmetic schemes. We will present their conjecture and the constructions necessary for its formulation in more detail. One may also consult Appendix A.6 for a schematic overview of the involved types of cohomology and their interconnections. Let d “ dim X and let n be any integer. • Flach and Morin have constructed perfect complexes RΓc pX , Rpnqq and RΓar,c pX , R̃pnqq “ RΓc pX , Rpnqq ‘ RΓc pX , Rpn ´ 1qq in the derived category of real vectorspaces, the former being the mapping cone of the Beilinson regulator map. Conjecturally, the vanishing orders of LpX , sq are determined 2 by the ranks of the resulting compact Arakelov cohomology groups: ords“n ζpX , sq “ ÿ i,n pX , R̃pnqq. p´1qi ¨ i ¨ dimR Har,c iPZ • Further, they assume the validity of the Artin-Verdier duality conjecture, i.e., the existence of a perfect pairing H ‚ pX , Z{mpnqq ˆ H 2d`1´‚ pX , Z{mpd ´ nqq ÝÑ Q{Z for positive integers m, to construct a perfect complex RΓW pX , Zpnqq out of the (completed) motivic cohomology RΓpX , Zpnqq of Bloch’s cycle complexes Zpnq. It contains all information of both the finitely generated as well as the cofinitely generated parts of motivic cohomology. They went on to define the compactly supported Weil-étale cohomology complex RΓW,c pX , Zpnqq via the triangle RΓW,c pX , Zpnqq ÝÑ RΓW pX , Zpnqq ÝÑ RΓW pX8 , Zpnqq ÝÑ (1) where the right-most complex is an integral version of Betti cohomology. • They have shown that the above complexes fit into a distinguished triangle pRΓddR pX {Zq{Filn qR r´1s ÝÑ RΓar,c pX , R̃pnqq ÝÑ RΓW,c pX , ZpnqqR ÝÑ (2) where RΓddR pX {Zq denotes the derived version of the de Rham cohomology of X . This motivates the definition of the fundamental line ∆pX , nq “ detZ RΓW,c pX , Zpnqq b detZ RΓddR pX {Zq{F n and gives rise to its trivialization – – λ8 pX , nq : R ÝÑ detR RΓar,c pX , R̃pnqq ÝÑ ∆pX , nqR . • They have defined a correction factor CpX , nq P Qˆ {t˘1u in terms of determinants of conjecturally distinguished triangles coming from p-adic Hodge Theory. CpX , nq is trivial for n ď 0. The leading Taylor coefficients ζ ˚ pX , nq are now conjectured to satisfy and hence be determined by λ8 pζ ˚ pX , nq´1 ¨ CpX , nq ¨ Zq “ ∆pX {Z, nq. Results and Layout. In this thesis we will explicate these conjectures for proper regular arithmetic surfaces X and show compatibility with the Birch and Swinnerton-Dyer conjecture. On the way we will establish decomposition results for various types of cohomology. We will interpret this as an indication for the existence of a decomposition of a hypothetical motive 3 hpX q into motivic degree components within the framework of a yet to be developed theory of mixed motives. In Chapter I we will prove Artin-Verdier duality for arithmetic surfaces X in the then remaining open case n “ 1. We will decompose X into a smooth open part on which duality is easy and a collection of bad fibers on which the duality statement follows from Saito’s work in [28]. The main result is Theorem 1.3. Artin-Verdier duality will be used as a computational tool in the remaining chapters. Chapters II and III are the core of this thesis. In the second chapter we will evaluate all cohomology groups introduced before, assuming the existence of a section s : S Ñ X , where S is the spectrum of the integer ring of a number field F for which there is a factorization π : X Ñ S of the structure map of X . A summary of the results is given in Appendix A.5. All computations will be organized around the main result Theorem 2.11 (and its various versions throughout Section 2.2) providing a decomposition p Rπ˚ ZpnqX » ZpnqS ‘ R1 π˚ ZpnqX r´1s ‘ Zpn ´ 1qS r´2s (3) in the derived category of abelian sheaves on S, and hence a decomposition of the associated cohomology groups. The analogue of (3) for torsion sheaves will be proven using Verdier duality and the six functor formalism for π and s. (3) will be an extension of it incorporating Geisser’s results on dualizing cycle complexes in [12]. A further insight will be the analysis of derived de Rham cohomology as given in Proposition 2.23. It will explain the occurrence of the Bloch-Kato conductor ApX q in the special value formulas later. In Chapter III we evaluate the trivialization factor coming from λ8 pX , nq (Theorem 3.13). Understanding λ8 pX , nq will amount to comparing the two integral structures of RΓW,c pX , ZpnqqR induced by the triangles (1) and (2). Note that (2) only conjecturally determines an integral lattice inside RΓW,c pX , ZpnqqR as follows: Flach and Morin have constructed a pairing Hc‚ pX , Rpnqq ˆ H 2d´‚ pX , Zpd ´ nqqR ÝÑ R (4) that they conjecture to be perfect and, moreover, to encode the Arakelov Intersection Pairing. This would endow the cohomology groups of the second term in (2) RΓar,c pX , R̃pnqq with a canonical integral structure coming from RΓpX , Zpnqq. Let m “ r ` 2s “ rF : Qs and write g for the genus of the generic fiber X of X . We further prove CpX , 1q “ 1 unconditionally (Theorem 3.19) and derive CpX , nq “ ppn ´ 1q! ¨ pn ´ 2q!qmpg´1q for n ě 2 assuming both the special value conjectures as well as the functional equation for 4 ζpX , sq (Theorem 3.24). We will combine this to the special value formulas $ ’ ˘p´1qi R2´n pSqR1´n pSq ś ` ’ ’ ’ 2pr´lpX qqn #Tor H i pX , Zp2´nqqcodiv ’ R2´n pX q ’ ’ 1ďiď4 ’ & 0 s 2 2 2r p2πq ζ ˚ pX , nq “ ? ¨ p#Tor Pic X q ¨ RpSq ’ p#µF q2 DF #XpX{F q¨ΩpX q RpX q ’ ’ ’ ’ ¯mpg´1q ´ ’ ’ ’ % pn´1q!¨pn´2q! ApX q1´n ¨ ζ ˚ pX , 2 ´ nq p2πq2pn´1q for n ď 0 for n “ 1 for n ě 2 Here the regulators Rn pX q and Rn pSq are determinants of the matrices describing the pairing (4) for X and S, and ΩpX q is the determinant of (a restriction of) the period isomorphism comparing Betti and de Rham cohomology. The case n ě 2 requires an additional technical assumption on X needed to evaluate its derived de Rham cohomology in terms of ApX q. We will conclude that Flach’s and Morin’s conjectures for surfaces X and n “ 1 are equivalent to the conjunction of the vanishing order part and the leading Taylor coefficient part of the Birch and Swinnerton-Dyer conjecture. The main result will be Theorem 3.27. 5 Chapter 1 Artin-Verdier duality for arithmetic surfaces Throughout this chapter let O be a number ring with fraction field F . Write S “ Spec O. X will denote a proper regular arithmetic scheme over O of pure dimension d – arithmetic, meaning that there is an integral, normal, excellent, flat map X Ñ Spec O of finite type with smooth generic fiber XF . We will write X for a proper regular arithmetic surface over O. Zpnq will denote the motivic cycle complexes as defined in Appendix A.1. Finally, let GR “ GalpC{Rq. In this chapter we combine work by Saito with Flach’s and Morin’s construction of the ArtinVerdier étale topos to generalize Artin-Verdier duality to arithmetic surfaces for coefficients given by Zp1q “ Gm r´1s. 1.1 The Artin-Verdier ètale topos and compact support cohomology Classical Artin-Verdier duality gives a pairing 3 H r pS, Fq ˆ Ext3´r S pF, Gm q ÝÑ H pS, Gm q “ Q{Z for all constructible étale sheaves F on S that is in general perfect only up to 2-torsion. Duality for the 2-torsion components needs further assumptions, e.g., that the underlying number field F is totally imaginary (cf. [21] Sec. 2). Geisser has generalized this result to arithmetic schemes X of any dimension (cf. [12] Thm. 7.8). Conjecturally, an analogous duality should hold for Bloch’s cycle complexes replacing 6 F. To treat the cases p ‰ 2 and p “ 2 uniformly Flach and Morin have constructed the Artin-Verdier étale topos X ét (cf. [8] App. A) which we will briefly review here. The Artin-Verdier étale topos X ét . Let Xét denote the étale topos of X and write X8 for both the quotient X pCq{GR itself and its associated topos of sheaves of set ShpX pCq{GR q interchangeably. The projection π : X pCq Ñ X8 extends to a morphism of topoi / π ShpGR , X pCqq ShpX8 q given by π ˚ pE Ñ X8 q “ E ˆ X pCq Ñ X pCq pπ˚ F qpU q “ F pπ ´1 U qGR , and X8 where E is the étalé space associated to a sheaf in X8 . Next, the functor that maps any étale covering U Ñ X to the GR -equivariant étalé space UpCq Ñ X pCq induces a morphism of topoi α : ShpGR , X pCqq Ñ Xét . We now define the Artin-Verdier étale topos as the topos X ét fitting into an Open-Closed-Decomposition φ u 8 Xét ÝÑ X ét ÐÝ X8 (1.1) such that u˚8 φ˚ – π˚ α˚ . Moreover, we may write π˚ as the composition p˚ ShpGR , X pCqq om pZ,´q / X , H / ShpGR , X8 q 8 where pp˚ FqpU q “ Fpπ ´1 U q and Hom pZ, ´q denotes the composition of the sheaf-hom functor inside ShpGR , X8 q with the forgetful functor. We summarize this in the diagram Xét e φ / X ét o α 7 u8 π ShpGR , X pCqq p˚ X O 8 om pZ,´q H / ShpGR , X8 q Next, let Pě0 Ñ Z Ñ 0 be the standard resolution of the constant sheaf Z with trivial GR -action. Write Γ˚ for the left-adjoint functor of the global sections functor for GR equivariant sheaves on X pCq. For any bounded below complex A‚ of sheaves in ShpGR , X q one has ż ‚ Rπ˚ A – Her ş Hom pΓ˚ Pě0 , p˚ A‚ q. denotes totalization. In analogy to Tate cohomology we define the Tate analogue Rp π of Rπ by replacing Pě0 with a complete resolution P‚ of Z: ż π˚ A‚ :“ Hom pΓ˚ P‚ , p˚ A‚ q.1 Rp 1 See [8] Sec. 6.4 for why Rp π is well-defined as a functor between derived categories and for further details. 7 For technical reasons the analogue ZpnqX of Bloch’s cycle complexes in the derived category of abelian sheaves on X ét is defined via the distinguished triangle ZpnqX ÝÑ Rφ˚ ZpnqX ÝÑ u8,˚ τ ąn Rp π˚ α˚ τ ě0 ZpnqX ÝÑ . However, if ZpnqX is cohomologically concentrated in degrees ď n then one has the more intuitive identity ZpnqX “ τ ďn Rφ˚ ZpnqX (cf. [8] Prop. 6.10). The cohomology of ZpnqX and ZpnqX only differ in 2-torsion and the precise difference will be addressed in the next chapter and in Appendix A.3. Milne’s étale cohomology with compact support. One formulation of Artin-Verdier duality uses compact support cohomology which we will briefly review here. Let U Ă S be an open subscheme. Write S8 for all infinite places of O and let Sfin “ SzU denote the set of all finite places of O not in U . For any abelian sheaf F on Uét Milne’s cohomology with p c pUét , Fq given via the distinguished compact support is the cohomology of the complex RΓ triangle p c pU, Fq ÝÑ RΓpU, Fq ÝÑ RΓ à à RΓpFv , fv˚ Fq ‘ vPSfin p v , f ˚ Fq ÝÑ . RΓpF v vPS8 p v , ´q Here fv : Spec Fv Ñ U is the canonical embedding of the closed point v in U and RΓpF denotes Tate cohomology of the Galois group of Fv (cf. [22] p.165ff). We extend this definition to schemes f : U Ñ U over U and abelian sheaves F on Uét by setting p c pUét , Fq :“ RΓ p c pUét , f˚ Fq. RΓ (1.2) p i pUét , Fq :“ H i pRΓ p c pUét , Fqq. We write H c This definition covers coefficients given by Zpnq for n ď 1 since these are cohomologically p c pXét , Zpnqq concentrated in one degree. It is consistent with the following definition of RΓ for general n. Let Rφp! ZpnqX be defined via the distinguished triangle Rφp! ZpnqX ÝÑ Rφ˚ ZpnqX ÝÑ u8,˚ Rp π˚ α˚ τ ě0 ZpnqX ÝÑ . p c pXét , Zpnqq :“ RΓpX ét , Rφp! ZpnqX q. Define RΓ 1.2 The Duality Statement Fix any integer n. The Artin-Verdier duality conjecture may be formulated as follows. Conjecture 1.1. AV(X ,n) Let m be a positive integer. There exists a product map ZpnqX {m bL Zpd ´ nqX {m ÝÑ ZpdqX {m 8 in the derived category of Artin-Verdier étale sheaves on X such that the induced pairing H i pX ét , Zpnq{mq ˆ H 2d`1´i pX ét , Zpd ´ nq{mq ÝÑ H 2d`1 pX ét , Zpdq{mq Ñ Q{Z is a perfect pairing of finite abelian groups integers i. Morin has shown that this is equivalent to the following formulation using Tate cohomology (cf. [8] Thm. 6.24). Conjecture 1.2. AV’(X ,n) For any positive integer m there exists a product map ZpnqX {m bL Zpd ´ nqX {m ÝÑ ZpdqX {m in the derived category of étale sheaves on X such that the induced pairing p i pXét , Zpnq{mq ˆ H 2d`1´i pXét , Zpd ´ nq{mq ÝÑ H p 2d`1 pXét , Zpdq{mq Ñ Q{Z H c c is a perfect pairing of finite abelian groups for all integers i. This conjecture is known for n ě d and n ď 0 and also for all n P Z if X is smooth over a number ring (cf. [8] Cor. 6.26, 6.27). We wish to prove AV(X ,n) for proper regular arithmetic surfaces X , i.e. for the case d “ 2. By the above, only the case n “ 1 remains to be shown. This will be the main result of this chapter. We formulate it for prime powers m “ pr . Theorem 1.3. (AV2) There exists a product map Zp1qX {pr bL Zp1qX {pr ÝÑ Zp2qX {pr (1.3) such that for all prime powers pr the induced pairing of cohomology groups p i pXét , Zp1q{pr q ˆ H 2d`1´i pXét , Zp1q{pr q ÝÑ H p 5 pXét , Zp2q{pr q Ñ Z{pr H c c (1.4) is a perfect pairing of finite pr -torsion groups for all integers i. Remark 1.4. The versions of Artin-Verdier duality in [8] conjecture the existence of a product map for the integral complexes ZpnqX bL Zpd ´ nqX ÝÑ ZpdqX that induce the aforementioned product maps of torsion complexes. For the pairing of cohomology groups this does not make a difference. We will construct a product map Zp1qX bL Zp1qX ÝÑ Zp2qX (1.5) under the additional assumption that Zp2qX satisfies the Beilinson-Soulé conjecture. (1.3) will be constructed unconditionally. 9 1.3 Lichtenbaum’s product map and Spiess’ complex Lichtenbaum’s complex Zp2, X q. Before Bloch defined the cycle complexes ZpnqX for general n ě 0 Lichtenbaum constructed a complex Zp2, X q using K-theory and proved that it satisfies many of the axioms for arithmetic cohomology complexes as proposed by Beilinson (cf. [20]). We review its definition. Let A be a regular noetherian ring. Write A1A “ Spec Arts and Z “ Spec Arts{tpt ´ 1q. a P A is an exceptional unit if both a, 1 ´ a P Aˆ . For a finite set B Ă A of exceptional units let ś YB “ Spec Arts{ bPB pt ´ bq. pYB qB forms an inverse system, so we may define 1 Cn,1 pAq “ lim Kn pA1A zYB , Zq and Cn,2 pAq “ lim Kn´1 pYB q ÝÑ ÝÑ B B (cf. [20] Def. 1.5). After gluing and sheafifying the presheaves Spec A ÞÑ Cn,i pAq for i “ 1, 2 X we obtain abelian sheaves C X n,1 and C n,2 on Xét . Definition 1.5. (cf. [20] Def. 2.1) Let Zp1, X q and Zp2, X q be the complexes in the derived category of abelian sheaves on Xét given by 0 1 X Zp1, X q “ rC X 1,1 Ñ C 1,2 s, 1 and 2 X Zp2, X q “ rC X 2,1 Ñ C 2,2 s. Lichtenbaum shows that K1 pX qr´1s is quasi-isomorphic to Zp1, X q (cf. [20] Prop. 2.4). As the Bloch cycle complex Zp1qX is quasi-isomorphic to K1 pX qr´1s we immediately have Zp1qX – Zp1, X q in the derived category. An analogous isomorphism for n “ 2 is only conjectured, but for arithmetic surfaces X partial results are known. Spiess’ complex K{X for arithmetic surfaces. Spiess constructed a complex K{X of abelian sheaves on Xét and proved a duality of type H i pX , Fq ˆ Ext6´i pF, K{X q ÝÑ Q{Z (1.6) for any constructible étale sheaf F on X (cf. [34] Theorem 2.2.2). We review the construction of K{X . Definition 1.6. Let « ff 1 2 à à à 3 à4 c2 K{X “ piξ q˚ C 2,1 pkpξqq ÝÑ piξ q˚ C 2,2 pkpξqq ÝÑ piη q˚ Gm ÝÑ i˚ Z , ξPX 0 ξPX 0 ηPX 1 xPX 2 where c2 arises from the composition of C2,2 pkpξqq Ñ K2 pkpξqq with the boundary map B 1 : K2 pkpξqq Ñ Gm from K-theory. 10 Note that there is a canonical map Zp2, X q Ñ K{X . Geisser has shown a duality for Zp2qX analogous to (1.6) (cf. [12] Thm. 8.7) which suggests that K{X and Zp2qX might coincide. Zhong has found a partial answer to this conjecture. Theorem 1.7. (Zhong, [38] Thm. 3.8) There is a map of complexes Zp2qX Ñ K{X in the derived category of etale sheaves on X that induces a quasi-isomorphism » τ ě1 Zp2qX ÝÑ K{X . In particular, [8] Conjecture 7.1 holds for arithmetic surfaces. Using Flach’s and Morin’s technical Lemma 7.7 in [8] we may also remove the truncation after passing to mapping cones. Corollary 1.8. Zp2qX is cohomologically concentrated in degrees ď 2. Moreover, for any prime power pr the complex Zp2qX {pr is cohomologically concentrated in degrees 0, 1, 2. Proof. Spiess remarks that K{X is concentrated in degrees 1, 2 (cf. [34] 1.6.2.(A1)). Consequently, [8] Conjecture 7.1 holds for Zp2qX and so [8] Lemma 7.7 applies. It yields for every prime p and its associated Open-Closed-Decomposition i j Xp ÝÑ X ÐÝ X r1{ps the distinguished triangle ˘ ` ď1 τ i˚ Zp1qXp {p r´2s ÝÑ Zp2qX {p ÝÑ τ ď2 Rj˚ µb2 ÝÑ, p where Xp :“ XFp . Since the left hand side is cohomologically concentrated in degree 3 it shows that H i pZp2qX {pq “ 0 for all i ă 0. Lichtenbaum’s product map. K-theory gives product maps Ki pAqbKj pT0 q Ñ Ki`j pT0 q, Ki pAq b Kj pT, T0 q Ñ Ki`j pT, T0 q for any closed immersion i : T0 ãÑ T of schemes of finite type over A. Therefore Cn,1 pAq, Cn,2 pAq have a K0 pAq-module structure and, moreover, there are maps Km pAq b Cn,1 pAq Ñ Cm`n,1 pAq. In particular, we obtain a product map K1 pX qr´1s b Zp1, X q ÝÑ Zp2, X q. (1.7) Recalling that K1 pXqr´1s – Zp1, Xq – Zp1qX and that the etale localizations pC i,1 pXqqx̄ “ Ci,1 pOX,x̄ q are flat for i “ 1, 2 (cf. [20] Prop. 2.5) we may rewrite the left-hand side as Zp1qX bL Zp1qX . Composition with Zhong’s isomorphism yields the product map » Zp1qX bL Zp1qX ÝÑ Zp2, X q ÝÑ K{X ÝÑ τ ě1 Zp2qX . (1.8) Taking mapping cones gives the pairing (1.3) as desired. When assuming Beilinson-Soulé τ ě1 Zp2qX » Zp2qX the above extends to the pairing (1.5) into the full Zp2qX . 11 1.4 A further construction of the p-torsion product map for arithmetic surfaces a la Sato Fix a prime power pr . We present an alternative construction of the p-torsion version of (1.3) Zp1qX {pr bL Zp1qX {pr ÝÑ Zp2qX {pr (1.9) that builds on a lifting argument by Sato (cf. [29] Prop. 4.2.6). For an arithmetic surface one can drop Sato’s normal crossing condition and use the explicit description of a boundary map from K-theory instead. This section should be understood as an extension of some of Sato’s results and constructions in [29] to arithmetic surfaces with not necessarily semistable fibers. An Open-Closed-Decomposition for ZpnqX . Let Z “ XFp denote the special fiber of X over p and write X r1{ps for its complement. We have the Open-Closed-Decomposition i j Z ÝÑ X ÐÝ X r1{ps. We know H i pZp2qX q “ 0 for i ą 2 (Corollary 1.8). Therefore – by Proposition A.5(ii) – Rj˚ j ˚ Zp2qX {pr “ Rj˚ µb2 pr , and we have the distinguished triangles i˚ ZZ {pr r´2s ÝÑ Zp1qX {pr ÝÑ τ ď1 Rj˚ µpr ÝÑ, ` ď1 ˘ τ i˚ Zp1qZ {pr r´2s ÝÑ Zp2qX {pr ÝÑ τ ď2 Rj˚ µb2 pr ÝÑ (1.10) (cf. [8] Lemma 7.7). [38] Thm. 1.1 provides the quasi-isomorphism « ff à à Z r 1 0 Zp1q {p » piη q˚ Wr Ωkpηq,log ÝÑ pix q˚ Wr Ωkpxq,log r´1s, ηPZ 0 (1.11) xPZ 1 i.e. Zp1qZ {pr is cohomologically concentrated in degrees 1 and 2. We wish to show that Zp1qX {pr bL Zp1qX {pr ÝÑ τ ď1 Rj˚ µpr bL τ ď1 Rj˚ µpr ÝÑ τ ď2 Rj˚ µb2 pr lifts to a duality pairing Zp1qX {pr bL Zp1qX {pr ÝÑ Zp2qX {pr . This will follow from the picture ` ď1 ˘ τ i˚ Zp1qZ {pr r´2s l / Zp2q{pr / τ ď2 Rj˚ µb2 pr O Hom“0 / τ ď1 i˚ Zp1qZ {pr r´1s 4 ` ˘ “0 Zp1qX {pr bL Zp1qX {pr (1.12) which we will verify now. Zp1qX {pr bL Zp1qX {pr is concentrated in degrees 0, 1, 2 while pτ ď1 i˚ Zp1qZ {pr qr´2s is concentrated in degree 3. So, there can in fact not be any non-trivial morphisms between them. Now it suffices to show the 12 Proposition 1.9. The composition ` ď1 ˘ Zp1qX {pr bL Zp1qX {pr ÝÑ τ ď2 Rj˚ µb2 i˚ Zp1qZ {pr r´1s pr ÝÑ τ on the right-hand side of (1.12) vanishes. X r L X r Proof. We use Sato’s notation Mnr “ i˚ Rn j˚ µbn pr . Observe that Zp1q {p b Zp1q {p Ñ ` ď1 ˘ τ i˚ Zp1qZ {pr r´1s can only be non-trivial in degree 2. The map on second cohomology sheaves is supported on Z and thus can be identified with H 1 Zp1qX {pr b H 1 Zp1qX {pr ÝÑ Mr2 ÝÑ H 1 pZp1qZ {pr q. Fix a geometric point z̄ of Z. Then, for A “ OX ,z̄ r1{ps, F “ Frac A, and any n ě 1 one has bn n n r pMrn qz̄ “ Hét pA, µbn pr q ãÑ Hét pF, µpr q “ Kn F {p . The inclusion is due to Gabber as mentioned in the proof of [2], Prop. 6.1. The last equality is only needed in the easy case n “ 1, 2 but it holds in general by the Rost-Voevodsky 1 pA, µ r q “ K A{pr . Theorem. Moreover, for n “ 1 the Kummer sequence shows that Hét p 1 Consequently — when writing Zz0 “ Z 0 X tzu for the collection of all non-closed points whose closure contains z — (1.10) yields ˜ ¸ pH 1 Zp1qX {pr qz “ Ker pMr1 qz Ñ à Z{pr ˜ “ Ker ηPZz0 à Z K1 A Ñ r p pr ηPZ 0 ¸ . z Also, ¨˜ pH 1 Zp1qZ {pr qz̄ “ Ker ˝ ¸ à piη q˚ Wr Ω1kpηq,log ηPZ 0 pix q˚ Wr Ω0kpxq,log xPZ 1 ‚ z̄ ¸ à “ Ker à ÝÑ z̄ ˜ ¸ ˛ ˜ kpηqˆ {pr ÝÑ Z{pr Ă ηPZz0 à kpηqˆ {pr . ηPZz0 So, it suffices to show that the composition ¸ ˜ ¸ ˜ à Z à Z K1 A K2 F ‘η Bη à K1 kpηq K1 A Ker Ñ b Ker Ñ ÝÑ ÝÝÝÝÑ (1.13) pr pr pr pr pr pr ηPZ 0 ηPZ 0 ηPZ 0 z z z Bη vanishes. The map K2 F {pr Ñ K1 kpηq{pr is the boundary map coming from K-theory with F being regarded as the fraction field of the DVR Aη . Let the corresponding valuation be called vη . The composition b Bη K1 F ˆ K1 F ÝÑ K2 F ÝÑ K1 kpηq admits the explicit formula avη pbq pa, bq ÞÑ p´1qvη paqvη pbq v paq bη 13 (cf [1] Prop. 4.5(e)). In particular, the image of a b b under (1.13) is trivial whenever Ď Now, since the map K1 F Ñ À 0 Z{pr is the tuple vη paq “ vη pbq “ 0 for all η P tzu. ηPZz À and any a P A is integral with respect to all v the composition (1.13) must be v Ě η ηPtzu η identically zero. The Gersten complex of logarithmic deRham-Witt sheaves of a variety X (which the righthand side of (1.11) is an example of) is known to be concentrated in one degree only if X is normal-crossing. For the one-dimensional curve Z the normal-crossing condition is not necessary. Proposition 1.10. The boundary map à à β: piη q˚ Wr Ω1kpηq,log ÝÑ pix q˚ Wr Ω0kpxq,log ηPZ 0 xPZ 1 is surjective, i.e. Zp1qZ {pr » Kerpβqr´1s. Proof. The Gersten complexes for Z and Z red are identical; thus we assume Z to be reduced. À The stalk of xPZ 1 pix q˚ Wr Ω0kpxq,log at a generic point η P Z 0 vanishes. It therefore suffices to consider the maps on stalks at a geometric point z̄ ãÑ z à 0 Wr Ω1kpηq,log ÝÑ Wr Ωkpxq,log βz : Ě ηPtzu for any closed point z P Z. Note that we now work over the algebraic closure k̄ of k since we are considering etale stalks. We may assume that Z is irreducible. Let η denote its generic point. By restricting to a suitable neighborhood of z we may assume Z to be affine. We are left to show that βz : kpηqˆ {pr ÝÑ Z{pr surjects. Let N Ñ Z be the normalization of Z and P1 , . . . , Pr P N the preimages of z. Each ř local ring ON,Pi is a DVR and hence comes with a valuation vi . One has βz “ i vi . So, it suffices to find a rational function f P kpN q “ kpZq which has a simple zero at precisely one Pi0 and is non-zero at all remaining Pi . This is easy for infinite base fields. Embed N Ă Ank . As k is infinite we can choose a hyperplane H Ă Ank passing through some Pi0 that does not contain any other Pj and that also does not contain the tangent direction of Z at x. The linear function f with Zpf q “ H will satisfy the above. Corollary 1.11. One has a distinguished triangle i˚ Zp1qZ {pr r´2s ÝÑ Zp2qX {pr ÝÑ τ ď2 Rj˚ µb2 pr ÝÑ i.e. [8] Conjecture 7.10 holds for arithmetic surfaces and n “ 2 and Sato’s complex Ir p2qX defined for normal-crossing X is quasi-isomorphic to Zp2qX and its construction in [29] Def. 4.2 can be carried out without the normal-crossing assumption. 14 1.5 Saito’s duality result and duality on the closed part For the remainder of this chapter we let X be a surface over the integer ring of a local field L with perfect residue field of characteristic p. We write Z for its special fiber and j : XL ãÑ X and i : Z ãÑ X for the canonical open and closed embeddings. Obviously, the constructions from last section can equally be carried out for X , i.e. we also have Spiess’ and Lichtenbaum’s complexes K {X and Zp2, X q. We will show (AV2) by proving dualities on the smooth, open part and the singular, closed part separately. This idea is based on Sato’s proof that Artin-Verdier duality in the global and local setting are equivalent (cf. [29] Sec. 10). Duality on the closed part has essentially been shown by Saito in [28]. We will need the analogue of [28] (4-1) for X and will briefly sketch Sato’s proof in this context. Theorem 1.12. (AV2l) Let X be a proper regular surface over the integer ring of a local field with special fiber Z. Write HZi pX , Zpnqq :“ H i pZ, Ri! Zpnqq. The product map Ri! Zp1qX bL i˚ Zp1qX ÝÑ Ri! Zp2qX in the derived category of abelian sheaves on Xét induced by (1.8) induces a perfect pairing trX ,Z HZi pX , Gm q ˆ H 4´i pZ, i˚ Gm q ÝÑ HZ6 pX , Zp2qX q ÝÝÝÑ Q{Z (1.14) of finitely generated abelian groups. Proof. We will replicate the computations in [28] Section 4 using the additional simplification that L cannot have a real embedding. Note that the trace map trX ,Z is Saito’s trace map constructed in [28] Thm. 3.1. The duality (1.14) can be written as RΓZ pX , Gm q˚ r´4s » RΓpZ, i˚ Gm q (1.15) in the derived category of abelian groups, where p´q˚ “ RHomp´, Q{Zq. To prove (1.15) we use the localization sequences à RΓx pX , Gm q ÝÑ RΓZ pX , Gm q ÝÑ xPZ0 à RΓx pZ, i˚ Gm q ÝÑ RΓpZ, i˚ Gm q ÝÑ xPZ0 and evaluate the enclosing complexes. à RΓη pX , Gm q ÝÑ (1.16) ηPZ1 à ηPZ1 RΓpOη , Gm q ÝÑ (1.17) 15 Computation of the distinguished triangle (1.16). We will now compute the terms RΓx pX, Gm q and RΓη pX, Gm q. For x P Z0 , η P Z1 write Dx “ Spec Ox zx and Kη for the fraction field of Oη . The inclusions txu ãÑ Ox and tηu ãÑ Oη give rise to the distinguished triangles RΓx pX , Gm q ÝÑ RΓpOx , Gm q Ñ RΓpDx , Gm q ÝÑ, RΓη pX , Gm q ÝÑ RΓpOη , Gm q Ñ RΓpKη , Gm q ÝÑ . Let kpxq, kpηq denote the residue fields of Ox , Oη respectively. It is well-known that τ ě1 RΓpOx , Gm q » τ ě1 RΓpkpxq, Gm q and τ ě1 RΓpOη , Gm q » τ ě1 RΓpkpηq, Gm q. As kpxq is finite, its cohomological dimension is 1. So, by Hilbert 90 and the fact that finite fields have trivial Brauer group RΓpOx , Gm q must be concentrated in degree 0 and we have RΓpOx , Gm q » Oxˆ r0s. Similarly, we obtain the cohomology sheaves of RΓpOη , Gm q and RΓpKη , Gm q. This may be written as the distinguished triangles Oηˆ ÝÑ RΓpOη , Gm q ÝÑ Br kpηqr´2s ÝÑ Kηˆ ÝÑ RΓpKη , Gm q ÝÑ Br Kη r´2s ÝÑ . Indeed, we do not have 2-torsion groups in higher degrees as Saito does in his proof since Kη is an extension of the local field L, i.e. it has no real embedding. As Oη is a DVR and the canonical map Br kpηq Ñ Br Kη injects, we obtain Zr´1s ÝÑ RΓη pX , Gm q ÝÑ Br Kη r´3s ÝÑ . Br kpηq Finally, since Ox zDx Ă Ox has codimension 2 one has H 0 pDx , Gm q “ ΓpOx , Gm q “ Oxˆ and H 1 pDx , Gm q “ Pic pDx q “ Pic pOx q “ 0. Also, H 2 pDx , Gm q – H 2 pOx , Gm q “ Br pOx q “ 0 where the first isomorphism is due to Grothendieck (cf. [14], III, Thm. 6.1(b)). The duality RΓpDx , Gm q » RΓpDx , Gm q˚ r´3s (cf. [27]) therefore proves that RΓx pX , Gm q » pOxˆ q˚ r´4s and the distinguished triangle (1.16) becomes À ηPZ1 Zr´1s À ˆ ˚ xPZ0 pOx q r´4s / RΓZ pX , Gm q  / RΓη pX , Gm q  Br Kη ηPZ1 Br kpηq r´3s À / 16 Taking duals and shifting 4 terms to the right gives ´ À ηPZ1 ¯ Br Kη ˚ r´1s Br kpηq (1.18)  / / RΓZ pX , Gm q˚ r´4s RΓη pX , Gm q˚ r´4s / À ˆ xPZ0 Ox  À ηPZ1 Q{Zr´3s Here we used finiteness of Z1 . Computation of the distinguished triangle (1.17). To compute RΓx pZ, i˚ Gm q we use the localization sequence RΓx pZ, i˚ Gm q ÝÑ RΓpOx , Gm q ÝÑ à RΓpOv , Gm q ÝÑ vPSx Here Sx “ h η Sx,η and Sx,η denotes the collection of all codimension 1 points of Spec Ox š lying over η. We write Sη “ š xPtηu0 Sx,η . Note that Sη can be viewed as the collection of all finite places v of the residue field kpηq. Also, let Ov denote the henselization of Oη at v and kpvq its residue field. We get ˜ ¸ à ˆ à Ov {Oxˆ r´1s ÝÑ RΓx pZ, i˚ Gm q ÝÑ Br kpvqr´3s ÝÑ . vPSx vPSx So, p1.17q becomes `À À xPZ0 ˆ vPSx Ov À  ˘ À {Oxˆ r´1s ˆ ηPZ1 Oη / RΓpZ, i˚ Gm q ˚ xPZ0 RΓx pZ, i Gm q À xPZ0 À  /  À ηPZ1 RΓpOη , Gm q À vPSx Br kpvqr´3s (1.19) /  ηPZ1 Br kpηqr´2s Fix η P Z1 . Then kpηq is either a number or a function field depending on whether η is a horizontal or vertical divisor of X . So, Class Field Theory gives us the short exact sequence 0 ÝÑ Br kpηq ÝÑ à Br kpvq ÝÑ Q{Z ÝÑ 0 vPSη and we see that (1.19) is isomorphic to the distinguished triangle ˜ ¸ à à ˆ à à ˆ Ov {Oxˆ r´1s ‘ Q{Zr´3s ÝÑ RΓpZ, i˚ Gm q ÝÑ Oη ÝÑ . xPZ0 vPSx ηPZ1 ηPZ1 17 We compare the above with the distinguished triangle (1.18). It remains to prove that the boundary maps δ3˚ ś ˆ xPZ0 Ox / ηPZ1 and à ´ À ¯ Br Kη ˚ Br kpηq ¸ ˜ δ31 à à xPZ0 vPSx Oηˆ ÝÝÝÝÑ ηPZ1 (1.20) Ovˆ {Oxˆ (1.21) have isomorphic kernels and isomorphic cokernels. This requires some preparation. Results from Kato’s higher local class field theory. Fix η P Z1 . Let I Ă Oη be an ideal. In analogy to the adeles and ideles of number fields we define , $ ˇ ˇ & ˆ where v P S . ˆ ˆ 1 ź ź ˇ a P O K K Ă x x v v ˇ v , AI,η “ “ pav qv P ˆ ˇ vPSη 1 ` IOvˆ % 1 ` IO for almost all v v ˇ vPSη ź Ovˆ Ă1 “ pOv {IOv qˆ Ă AI,η . ˆ vPSη 1 ` IOv vPSη We have diagonal embeddings of Kηˆ , Oηˆ into AI,η , II,η respectively. We define II,η “ ź Ă1 CI,η “ AI,η {Kηˆ , 0 CI,η “ II,η {Oηˆ . The existence of a perfect pairing H i pKv , Gm q ˆ H 2´i pKv , Gm q Ñ H 4 pKv , Zp2qq – Q{Z. is well-known. The induced product x´, ´yv : Br pKv q ˆ Kvˆ ÝÑ Q{Z has the property that for each x P Br pKv q the map xx, ´yv has non-vanishing kernel. Br Ov can be characterized as the subgroup of Br Kv consisting of those elements x for which xx, ´yv vanishes on Ovˆ (cf. [29] Thm. 2.9). Consequently there is a map ˙ ˆ Kvˆ , Q{Z . Br pKv q ÝÑ lim Hom ÝÑ 1 ` IOvˆ IĂOv À Using the embedding Br pKη q ãÑ vPSη Br pKv q we get a map ˙ ˆ à Kvˆ , Q{Z “ lim HompAη,I , Q{Zq. Br pKη q ÝÑ lim Hom ÝÑ ÝÑ 1 ` IOvˆ IĂOη IĂOη vPSη Kato has shown that the above map factors through and surjects onto lim HompCI,η , Q{Zq ÝÑ IĂOη (cf. [29] Thm.2.13), i.e. we have an isomorphism Br pKη q – lim HompCI,η , Q{Zq. ÝÑ IĂOη 18 Using the characterization of Br Oη “ Br kpηq from before we obtain ` 0 ˘ Br pKη q – lim Hom CI,η , Q{Z . ÝÑ Br kpηq IĂO (1.22) η Comparison of kernels and cokernels. Note that Oxˆ “ lim pOx {IOx qˆ ÐÝ IĂOX since Ox is a localization of OX . Analogous statements hold for Oη , Ov . So — since Oˆ ˆ ˙ ˙ ź 1 ˆ Ov ˙ˆ à Ą à Br Kη ˚ à Oη ˆ 0 lim CI,η “ lim “ ÐÝ ÐÝ Br kpηq IOv IOη ηPZ1 ηPZ1 IĂO IĂO ηPZ1 vPS η X η ˚ and by (1.22) — we may rewrite the maps (1.20) and (1.21) as inverse limits δ3˚ “ lim δ3,I ÐÝ IĂOX 1 1 δ3 “ lim δ3,I with ÐÝ IĂOX ˚ δ3,I : ź ˆ Ox ˙ˆ xPZ0 IOx ÝÑ ź1 à Ą ηPZ1 vPSη and ˆ 1 δ3,I : à ηPZ1 Oη IOη ˆ ˜ ˙ˆ ÝÑ ˆ à à xPZ0 vPSx Ov IOv Ov IOv ˙ˆ Oˆ Oη IOη ˙ˆ ¸ Oˆ ˙ˆ Ox IOx ˙ˆ . ˚ and δ 1 δ3,I 3,I have isomorphic kernels and isomorphic cokernels already. Indeed, when writing ˆ ˙ ź à ˆ Ov ˙ˆ ź ˆ Ox ˙ˆ ź ˆ Oη ˙ˆ à ź Ov ˆ A“ “ , A0 “ , A1 “ IOv IOv IOx IOη ηPZ1 vPS xPZ vPSx xPZ ηPZ 0 η 0 1 ˚ , δ1 we have canonical embeddings A0 , A1 ãÑ A and the kernels of δ3,I 3,I are isomorphic to A0 X A1 . Similarly, for the direct sum analogues ˆ ˆ ˙ ˆ ˙ ˙ à à Ov ˆ à à Ov ˆ à Ox ˆ “ , A10 “ , A1 “ IOv IOv IOx xPZ0 vPSx ηPZ1 vPSη xPZ0 ˆ A11 “ à ηPZ1 Oη IOη ˙ˆ ś1 ˚ , δ1 1 1 1 the definition of Ă ensures that the cokernels of δ3,I 3,I are isomorphic to A {pA0 ` A1 q. This completes the proof of Theorem (AV2l). 1.6 Duality on the open part Theorem 1.13. (AV2s) Fix an integer n and a prime power pr . Let U Ă S be an open subscheme and let f : U Ñ U denote a smooth scheme over U of dimension d. There is a duality in the derived category of abelian groups bpd´nq ˚ p c pU, µbn RΓ pr qr2d ` 1s » RΓpU, µpr q , 19 where p´q˚ “ RHomp´, Q{Zq denotes the Pontryagin-dual. In particular, for any integer i we have a perfect pairing bpd´nq 2d`1´i r p i pU, µbn p 2d`1 pU, µbd H pU, µpr q ÝÑ H c c pr q ˆ H pr q – Z{p . Proof. We will use the structure map f : U Ñ U to reduce the above statement to usual Artin-Verdier duality for number fields. The key ingredient will be the cohomological purity result [23] XVI. Thm. 3.7, which itself is an application of the Smooth Base Change Theorem [23] XVI. Thm. 1.1. Recall that p i pU, Fq ˆ Ext3´i pF, Gm q ÝÑ H 3 pU, Gm q “ Q{Z H c U is a perfect pairing for any constructible sheaf of abelian groups F over U . For sheaves F killed by pr this gives the quasi-isomorphism p c pU, Fqr`3s » R Hom U pF, µpr q˚ . RΓ We choose F “ Rf˚ pµbn pr qU . This yields bn p c pU, µbn p RΓ pr q » RΓc pU, Rf˚ pµpr qU q ` ˘˚ r´3s » R Hom U Rf˚ pµbn pr qU , µpr ¯ ´ ˚ pV q ! r´3s » R Hom U µbn pr , Rf pµpr qU ¯˚ ´ pSq ˚ bd r´3s » R Hom U µbn pr , Rf µpr r2d ´ 2s ´ ¯˚ ˚ bd » RΓ U, RHom U pµbn r´2d ´ 1s pr , Rf µpr q (1.23) bpd´nq ˚ » RΓpU, µpr q r´2d ´ 1s. For (V) we used the Verdier duality adjunction Rf˚ $ Rf ! (see also Theorem 2.3). (S) follows from Smooth Base Change applied to f : U Ñ U . Note that this quasi-isomorphism is why we require U to be smooth. 1.7 Proof of the global duality statement We first rewrite (AV2l) in terms of p-torsion sheaves. We will then combine it with (AV2s) to prove global Artin-Verdier duality. Proposition 1.14. Keep the notation from Theorem (AV2l) and let pr be a prime power. For each integer i there is a perfect pairing HZi pX , Zp1qX {pr q ˆ H 5´i pZ, i˚ Zp1qX {pr q ÝÑ HZ5 pX , Zp2qX {pr q ÝÑ Q{Z. (1.24) 20 Proof. Applying Ri! to the product map of pr -mapping cones (1.9) gives Ri! Zp1qX {pr bL i˚ Zp1qX {pr ÝÑ Ri! Zp2qX {pr which induces the pairing (1.24). We show that it is perfect by means of a Five Lemma argument. Consider the long exact sequence associated to ZpnqX Ñ ZpnqX Ñ ZpnqX {pr Ñ as well as its dual under p´q˚ “ RHomp´, Z{pr q. These — together with the product maps (1.14) and (1.24) — fit into the commutative diagram HZi pX , Zp1qX q – p1.14q  H 6´i pZ, i˚ Zp1qX q˚ / H i pX , Zp1qX q / H i pX , Zp1qX {pr q Z Z – p1.14q  / H 6´i pZ, i˚ Zp1qX q˚ p1.24q  / H 5´i pZ, i˚ Zp1qX {pr q˚ / H i`1 pX , Zp1qX q Z – p1.14q  / H 5´i pZ, i˚ Zp1qX q˚ / H i`1 pX , Zp1qX q Z – p1.14q  / H 5´i pZ, i˚ Zp1qX q˚ The outer vertical being isomorphisms is precisely the local duality result (AV2l). So, by the Five Lemma, the middle arrow represents a perfect pairing too. Proof of (AV2). Let Z be the collection of all special fibers of X that are singular or above š p, i.e. Z “ pPSbad Zp where B is the (finite) union of all primes of O where X has bad reduction with all primes above p, and where Zp “ X ˆ Spec kppq. Write U “ X zZ. The i j Open-Closed-Decomposition Z ãÑ X Ðâ U gives rise to the distinguished triangles j! µpr ÝÑ Zp1qX {pr ÝÑ i˚ i˚ Zp1qX {pr ÝÑ (1.25) i˚ Ri! Zp1qX {pr ÝÑ Zp1qX {pr ÝÑ Rj˚ µpr ÝÑ (1.26) Theorem (AV2s) gives us a perfect pairing on the smooth part r p 5 pU, µb2 p i pU, µpr q ˆ H 5´i pU, µpr q ÝÑ H H c c pr q – Z{p . (1.27) Consider the long exact sequence on compact support cohomology of the triangle (1.25) as well as the long exact sequence of (1.26). They fit into one commutative diagram as follows: H i´1 pZ, i˚ Zp1qX {pr q  p1.24q HZ6´i pX , Zp1qX {pr q˚ p i pU, µpr q /H c – p1.27q  / H 5´i pU, µpr q˚ p i pX , Zp1qX {pr q /H c  p1.4q / H 5´i pX , Zp1qX {pr q˚ / H i pZ, i˚ Zp1qX {pr q  p1.24q / H 5´i pX , Zp1qX {pr q˚ Z p i`1 pU, µpr q /H c – p1.27q  / H 4´i pU, µpr q˚ The vertical arrows are induced by the pairings (1.24), (1.27), and (1.4) from (AV2) as indicated. We explore the map H i pZ, i˚ Zp1qX {pr q ÝÑ HZ5´i pX , Zp1qX {pr q˚ (1.28) š in more detail. Write XB “ p Xp where Xp “ X ˆ Spec Op . One may regard i as an embedding of Z into XB . The Proper Base Change Theorem then gives H i pZ, i˚ Zp1qX {pr q “ H i pXB , Zp1qXB {pr q and we may write (1.28) as H i pXB , Zp1qXB {pr q ÝÑ HZ5´i pXB , Zp1qXB {pr q˚ . 21 So, this map is in fact the direct sum of (1.24) for each Zp ãÑ Xp with p P B and hence an isomorphism. By the Five Lemma, the middle vertical arrow must be an isomorphism too, i.e. (1.4) is a perfect pairing. 22 Chapter 2 Motivic decompositions of cohomology for arithmetic surfaces Let X denote a regular arithmetic surface with proper structure map π 1 : X Ñ Spec Z. Let O be the maximum number ring π 1 factors through. Unless explicitly stated otherwise, we write S “ Spec O and let π : X Ñ S denote the corresponding map into S. Given a point ś x P X or p P S we write κpxq and κppq for their residue fields. We will write p to denote an infinite product over all finite primes of O. Write F for the fraction field of O. Let r and s be the number of real and complex embeddings of F respectively and write m “ r ` 2s for its dimension over Q. If not explicitly stated otherwise, X will denote the generic fiber XF of X . Let g be the genus of X. Also, let n P Z and let n be 0 or 1 depending on whether n is even or odd. In this chapter we will define and partially compute the cohomology groups occuring in the conjectures [8] Conj. 5.10, 5.11 for X . The underlying theme will be that all cohomology groups will decompose into a direct sum of h i -parts for i “ 0, 1, 2 — analogously to a decomposition of ζpX , sq “ ź p i LpH i pX q, sqp´1q “ i“0,1,2 ζF psqζF ps ´ 1q p LpH 1 pX q, sq p into an alternating product of adjusted L-functions LpH i pX q, sq for i “ 0, 1, 2. We expect this to follow in generality from the existence of a direct sum decomposition of Rπ˚ ZpnqX p into perverse degree components Ri π˚ ZpnqX : à p i p Rπ˚ ZpnqX » R π˚ ZpnqX r´is » ZpnqS ‘ R1 π˚ ZpnqX r´1s ‘ Zpn ´ 1qS r´2s. i“0,1,2 We will prove such a decomposition under the assumption that there is a section s : S Ñ X of π satisfying a further technical conjecture for higher twists n ě 2. 23 2.1 L- and ζ-functions associated to arithmetic surfaces In this section we relate ζpX , sq to the Hasse-Weil ζ-function ζHW pX, sq and review the decomposition of ζpX , sq into its 0-, 1-, and 2-part. For any prime p of O let l “ lp denote a rational prime not divisible by p. Moreover, write Xp :“ Xκppq . The arithmetic ζ-function associated to any proper arithmetic surface X is defined as ź 1 “ ζ pXp {κppq, sq´1 ´s 1 ´ #κpxq p xPX0 źź ` ˘p´1qm`1 det id ´ N p´s Frobp | H m pXp , Ql q “ . ζpX , sq “ ź p mPZ The last equality above is a consequence of the Weil conjectures. The Hasse-Weil ζ-function on the other hand is an object associated to the generic fiber X “ XF and is thus independent of the integral model X of X. ζHW pX, sq is defined as an alternating product of Hasse-Weil L-functions which in turn only depend on the étale cohomology groups H m pXQ , Ql q. Concretely, one defines ζHW pX, sq :“ ź LpH m pXq, sqp´1q m`1 mPZ :“ źź ` ˘p´1qm`1 det id ´ N p´s Frobp | H m pX bF F , Ql qIp , (2.1) mPZ p where Ip Ă GalpQ{F q denotes the inertia group of some prime p of Q lying over p. Note that ζ and ζHW should be thought of as associated to the scheme X or, equivalently, to X regarded as an arithmetic surface over Z (or its generic fiber X regarded as a curve over Q), and not to arithmetic surfaces over a general number ring. However, we will make frequent use of the map π : X Ñ S since many objects associated to X such as its motivic cohomology groups can be best expressed in terms of S. One knows that H m pXp , Ql q – H m pXQ , Ql qIp for good reduction primes p. So, the difference between ζpX , sq and ζHW pX, sq lies in the bad reduction fibers of X only. Bloch worked out explicitly the difference between the above étale cohomology groups and arrived at the following result (cf. [3] Lem. 1.2 and comments). Proposition/Definition 2.1. Let Mp denote the Ql -vectorspace freely generated by the irreducible components of Xp . One has H i pXQ , Ql qIp “ H i pXp , Ql q for i “ 0, 1. For i “ 2 p there is an exact sequence of Z-modules 0 ÝÑ Mp {Ql p´1q ÝÑ H 2 pXp , Ql q ÝÑ H 2 pXQ , Ql qIp ÝÑ 0 24 and, moreover, H 2 pXp , Ql q – Mp˚ p´1q. We define ź ΠpX , sq :“ ` ˘ det id ´ N p´s Frobp | Mp {Ql p´1q . p bad The short exact sequence shows ζpX , sq “ ζHW pX, sq ¨ ΠpX , sq´1 . (2.2) Since H 0 pXp , Ql q “ Ql and H 2 pXQ , Ql q “ Ql p´1q we may simplify (2.1) to ζpX , sq “ ζF psqζF ps ´ 1q ζF psqζF ps ´ 1q “: p . 1 ΠpX , sqLpH pXq, sq LpH 1 pX q, sq (2.3) For future reference, we analyze the special values of ΠpX , sq. Lemma 2.2. For any prime p of F write dppq “ dimQl H 2 pXp , Ql q for the number of irreducible components C1 , . . . , Cdppq of Xp . Further, for each 1 ď j ď dppq let nj “ nj ppq be the number of irreducible components Cj decomposes into in Xp . (i) For any integer n ‰ 1 one has ź ˆ Π pX , nq “ ΠpX , nq “ ˘ 1´ ˚ p bad 1 N pn´1 ˙´1 dppq źˆ 1´ j“1 1 N ppn´1qnj ppq ˙ . (ii) For n “ 1 one has ords“1 ΠpX , sq “ ÿ pdppq ´ 1q (2.4) p bad and the leading Taylor coefficient Π˚ pX , 1q equals Π˚ pX , 1q “ ˘ ź p bad plog N pqdppq´1 dppq ź nj ppq. j“1 Proof. Fix a bad prime p and write d “ dppq and nj “ nj ppq. Let Mpj denote the subspace of Mp generated by the irreducible components of Cj in Xp . Write Pjp and P p for the characteristic polynomials for the action of Frobp on Mpj and Mp {Ql respectively. Frobip cyclically permutes the components generating each Mpj since the Frobp -orbit of any irreducible component of Cj in Xp must be defined over κppq by Galois-descent, and hence equal Cj . Therefore P p pT q “ d d ¯ 1 ź p 1 ź ´ nj ppq Pj pT q “ ˘ T ´1 . T ´ 1 j“1 T ´ 1 j“1 25 Write N “ dimQl H 2 pXp , Ql q “ ř s´1 . One has j nj . Let s P C and write x “ xp “ N p d x ź xnj ppq ´ 1 x x ´ 1 j“1 xnj ppq ˆ ˙ ˙ d ˆ 1 ´1 ź 1 1 ´ n ppq . “˘ 1´ x x j j“1 ˇ ` ˘ det 1 ´ N p´s`1 Frobp ˇ Mp {Ql “ 1 P p pxq “ ˘ N ´1 Choosing s “ n for n ‰ 1 proves part (i) and evaluating the vanishing order of the above at s “ 1 yields (2.4). Now let yp “ ´ log N p. Then x1p “ eyp ps´1q and ˜ Π˚ pX , 1q “ ˘ lim sÑ1 ź p bad 1 ´ eyp ps´1q s´1 ¸´1 dppq ź 1 ´ eyp nj ppqps´1q j“1 s´1 “˘ ź ź 1 dppq yp nj ppq. y p bad p j“1 This finishes the proof of part (ii). 2.2 Motivic decompositions of push-forward sheaves in the presence of a section In this section we will write π : X Ñ S for a variety of structure maps, as specified in the proceeding paragraphs. We will derive direct sum decompositions of the kind Rπ˚ F pnqX » F pnqS ‘ R1 π˚ F pnqX r´1s ‘ F pn ´ 1qS r´2s, p where F represents locally constant sheaves with Galois twist or motivic cycle complexes on the étale sites of X and S respectively. These results will be motivated by the theory of motives and hence referred to as motivic decompositions. We will also write Rπ˚ F pnqX » À p i p X i“0,1,2 R π˚ F pnq r´is. The notation R should indicate the expectation that those decompositions are reflections of a broader, yet to be developed theoretical framework which endows derived categories of motivic sheaves with perverse t-structures. The proof strategies are centered on an application of Verdier duality. It will render the remaining part of the proof an exercise in the six functor formalism for π and s. We will also provide more explicit descriptions of the involved projection and inclusion morphisms p when possible and elaborate on the difference between Ri and Ri . 2.2.1 Verdier Duality and and Cohomological Purity Verdier Duality. For any scheme X let DpXét q denote the derived category of abelian torsion sheaves on Xét . We recall the sheaf theoretic generalization of Poincaré Duality. 26 Theorem 2.3. (Verdier Duality, [23] Exp. XVIII, Thm 3.1.4) Let f : X Ñ Y be a separated, quasi-compact morphism of schemes. There is a functor f ! : DpYét q Ñ DpXét q such that for all torsion sheaves F on Xét and G on Yét one has the quasi-isomorphism Rf˚ R Hom X pF, f ! Gq » R Hom Y pRf! F, Gq. (2.5) In other words, there is an adjunction of functors Rf! $ f ! between the derived categories of torsion sheaves on Xét and Yét . f ! generalizes the derived exceptional inverse image functor associated to closed immersions. However, in general f ! cannot be expressed as the derived functor of any functor of sheaves. The counit trf of the adjunction Rf! $ f ! is called the trace map. The first part of the proof of Theorem 2.3 is the construction of the trace map. It is then used to define the map (2.5) and one subsequently shows that it is in fact a quasi-isomorphism. If f is smooth of relative dimension d (i.e. dim X “ dim Y ` d), one knows that Rf! f ! F is cohomologically concentrated in degrees r0, 2ds for any torsion sheaf F on Xét and trf pFq factors through the top degree: Trf trf pFq : Rf! f ! F ÝÝÝÝÑ R2d f! f ! F ÝÝÝÝÑ F. (2.6) Cohomological Purity. For the rest of this section fix a prime p and let Λ denote a finite p-torsion group. For any scheme X we write ΛX for the constant torsion sheaf with global sections equal to Λ. For any p-torsion sheaf F we write Fpnq “ F b µbn p8 . We will need the following result that had originally been conjectured by Grothendieck and that was proved in full generality by Gabber (cf. [10]). Theorem 2.4. (Cohomological Purity) Let i : Z ãÑ U be a closed immersion of regular noetherian schemes of pure codimension d. Suppose p is invertible on U . Then # ΛZ p´dq if r “ 2d, R r i! ΛU – 0 otherwise. (2.7) This result admits an extension to the functor f ! between derived categories. Proposition 2.5. Let f : X Ñ Y be a smooth morphism of relative dimension d. Also, let F be a complex of p-torsion sheaves on Xét and suppose p is invertible on Y . Then f ! F » f ˚ Fpdqr2ds. Let π : X Ñ S be a projective morphism of schemes with X being regular and of relative dimension d over S. Note that this includes the later most relevant case where d “ 1 and 27 π is any proper morphism since projectiveness follows then from a result by Lichtenbaum (cf. [19] Thm. 2.8). We assume throughout that p is not zero in OS . We then have open closed decompositions as in the diagram X r1{ps  jp Xp  jp1 (2.8) πp π πr1{ps Sr1{ps ip / X o i1p /So  Sp Lemma 2.6. In the derived category of torsion sheaves on Sr1{psét , one has πr1{ps! ΛSr1{ps » ΛX r1{ps pdqr2ds. Proof. πr1{ps is projective and hence factors through some PN Sr1{ps as X r1{ps / PN i Sr1{ps π $  Π Sr1{ps where Π is smooth and i is a closed embedding of regular schemes. So, by Theorem 2.4 and Proposition 2.5, PN πr1{ps! ΛSr1{ps “ Ri! Π! ΛSr1{ps » Ri! Λ Sr1{ps pN qr2N s » ΛX r1{ps pdqr2ds. l Corollary 2.7. Write p´q_ “ R Hom p´, Qp {Zp q. For any p-torsion sheaf F on X r1{psét one has pRπr1{ps˚ Fq_ » Rπr1{ps˚ F _ pdqr2ds. (2.9) Proof. This is Verdier duality for f “ πr1{ps and G “ ΛSr1{ps . Here we have used πr1{ps! “ πr1{ps˚ which holds since πr1{ps is proper. 2.2.2 A motivic decomposition for push-forwards of constant torsion sheaves 1 ΛSr1{ps . Note that if p is To ease notation we write ΛX ,p “ jp,! ΛX r1{ps and ΛS,p “ jp,! invertible on S then one gets the constant sheaves ΛX ,p “ ΛX and ΛS,p “ ΛS back. From now on, we assume S to be a connected at most one-dimensional regular scheme such that all closed points have perfect residue fields and such that all remaining points have residue fields of characteristic 0. Although we will need the result below only for d “ 1 we formulate it for general d as it will make no difference for the proof. 28 Theorem 2.8. Suppose π : X Ñ S has a section s : S Ñ X . Then the sheaves ΛS,p pnq and ΛS,p pn ´ dqr´2ds split off as direct summands of the complex Rπ˚ ΛX ,p pnq. If d “ 1 we will write R1 π˚ ΛX p pnqr´1s for the remaining summand, i.e. we will have the canonical p decomposition Rπ˚ ΛX ,p pnq » ΛS,p pnq ‘ R1 π˚ ΛX ,p pnqr´1s ‘ ΛS,p pn ´ 1qr´2s. p (2.10) Proof. It is enough to prove the claim for the restriction πr1{ps : X r1{ps Ñ Sr1{ps to the open part, i.e. to show that ΛSr1{ps pnq and ΛSr1{ps pn ´ dqr´2ds split off as direct summands 1 will then reproduce the original claim of Rπr1{ps˚ ΛX r1{ps pnq. In fact, an application of jp,! since 1 jp,! Rπr1{ps˚ ΛX r1{ps pnq “ Rpjp1 πr1{psq! ΛX r1{ps pnq “ Rπ! jp,! ΛX r1{ps pnq “ Rπ˚ ΛX ,p pnq. Therefore we may assume that p is invertible on S. We will use the adjunctions π ˚ $ π˚ and s˚ $ s˚ $ s! to construct maps ϕ0 ψ0 ϕ ψ ΛS pnq ÝÑ Rπ˚ ΛX pnq ÝÑ ΛS pnq, 2d 2d ΛS pn ´ dqr´2ds ÝÑ Rπ˚ ΛX pnq ÝÑ ΛS pn ´ dqr´2ds (2.11) that compose as ˜ ψ0 ¸ ψ2d ˜ ´ ˝ ¯ ϕ0 ϕ2d “ id 0 0 id ¸ , thereby proving the proposition. The existence of ϕ0 , ψ0 is a formal consequence of functoriality of the push-forward π˚ s˚ “ pπsq˚ “ id and exactness of π ˚ , s˚ , s˚ . Using s˚ ΛX “ ΛS and π ˚ ΛS “ ΛX we define ϕ0 : ΛS pnq ÝÑ Rπ˚ π ˚ ΛS pnq “ Rπ˚ ΛX pnq, ψ0 : Rπ˚ ΛX pnq Ñ Rπ˚ s˚ s˚ ΛX pnq “ s˚ ΛX pnq “ ΛS pnq. Consider the shift of the above maps for an pd ´ nq-twist by 2d degrees: ϕ0 r2ds ψ0 r2ds ΛS pd ´ nqr2ds ÝÝÝÝÑ Rπ˚ ΛX pd ´ nqr2ds ÝÝÝÝÑ ΛS pd ´ nqr2ds. Apply p´q_ “ R Hom S p´, Qp {Zp q. Corollary 2.7 shows that one obtains ϕ_ 0 r´2ds ψ0_ r´2ds ΛS pn ´ dqr´2ds ÐÝÝÝÝÝÝÝ Rπ˚ ΛX pnq ÐÝÝÝÝÝÝÝ ΛS pn ´ dqr´2ds. We let ϕ2d “ ψ0_ r´2ds and ψ2d “ ϕ_ 0 r´2ds. ψ0 ϕ0 “ id is clear since both maps are adjoints of identity maps. Therefore, also ψ2d ϕ2d “ pψ0 ϕ0 q_ r´2ds “ 0. Next, one has ψ2d ϕ0 “ 0 for degree reasons. More precisely, ϕ0 is 29 the inclusion of the lowest degree ΛS pnq “ R0 π˚ ΛX pnq into Rπ˚ ΛX pnq while ψ2d factors through the top degree τ ě2d Rπ˚ ΛX pnq – R2d π˚ ΛX pnq. Similarly, we must have ψ0 ϕ2d “ 0 since ϕ2d maps ΛS pn ´ dqr´2ds into the top degree of Rπ˚ ΛX pnq while ψ0 is trivial in degrees ą 0. This completes the proof. Remark 2.9. The construction of ϕ0 , ψ0 did not require ΛX to be torsion or p to be invertible on S. So, the above proof more generally shows that AS splits off as a direct summand of Rπ˚ AX for the constant sheaf AX associated to any abelian group A. Since both X and S are connected one has π˚ AX “ AS and consequently Rπ˚ AX » AS ‘ τ ě1 Rπ˚ AX . Remark 2.10. If π : X Ñ S is smooth and proper, and S the spectrum of a field of characteristic unequal to p the above decomposition is well-known and the motivic components p i R π˚ ΛX pnq coincide with the cohomological components Ri π˚ ΛX pnq for degrees i “ 0, 2d. In particular, if d “ 1, they are identical for all degrees, i.e. one then has Rπ˚ ΛX pnq » R0 π˚ ΛX pnq ‘ R1 π˚ ΛX pnqr´1s ‘ R2 π˚ ΛX pnqr´2s. This also follows from direct computations when observing that (2.11) may be rewritten as ϕ1 2d π ΛS pn ´ dqr´2ds ÝÑ R2d π˚ ΛX pnq ÝÑ ΛS pn ´ dqr´2ds, Tr (2.12) where Trπ is the trace map from Poincaré duality for etalé cohomology and ϕ12d is obtained from applying Rπ˚ to the adjoint of a cohomological purity isomorphism Rs! ΛS pn´dqr´2ds » ΛX pnq. We omit the details. For a general proper regular arithmetic surface π : X Ñ S the maps in (2.12) are not necessarily isomorphisms. This suggests that the motivic decomposition of Rπ˚ ΛX pnq arises from the (standard) cohomological degree components Ri π˚ ΛX pnq as follows: Split R2 π˚ ΛX pnq into a component dual to R0 π˚ ΛX pnq and into another component describing the obstruction of X from being smooth and then regroup the latter to the motivic degree p 1 part. This pattern is familiar from (2.3) and the contained definition LpH 1 pX q, sq :“ ΠpX , sqLpH 1 pXq, sq showing that the standard Hasse-Weil function ζHW pX, sq differs from the motivic function ζpX , sq only by additional terms in motivic degree 1 that are characterized entirely by the bad fibers of X . 2.2.3 A motivic decomposition for Rπ˚ Zpnq The fundamental insight for all following computations of motivic cohomology will be that Theorem 2.8 has an analogue for Bloch’s cycle complexes. We will need a technical preparation. 30 For any map f : X Ñ Y between arithmetic schemes and any étale neighborhood V Ñ Y define ZYn pV, jqf :“ tZ P ZYn pV, jq | pf ˆ id∆j q´1 pZq intersects all faces properlyu. For a section s : S Ñ X of π and n P Z we introduce the technical assumption n p´, 2n ´ ‚qs ãÑ Z n p´, 2n ´ ‚q gives The inclusion of simplicial structures ZX X rise to a quasi-isomorphism of associated derived complexes. In other words, FPB(s,n) n DKpZX p´, 2n ´ ‚qs q » ZpnqX . FPB(s,n) ensures the existence of a functorial pull-back morphism s˚ ZpnqX Ñ ZpnqS whose adjoint ZpnqX Ñ s˚ ZpnqS is given over an étale neighborhood U Ñ X by the map Z n pU, jqs ÝÑ Z n ps´1 U, jq $ ’ ’ & ps ˆ id j q´1 Z ∆ Z ÞÑ ’ ’ % 0 if for all closed points x P ∆j : Z X pX ˆ txuq has codim 0 in X ˆ txu, or is a finite union of vertical divisors of X ˆ txu (2.13) otherwise Note that for n ď 1 one has a morphism s˚ ZpnqX Ñ ZpnqS that is functorial in s unconditionally since this is well-known for the sheaves Qp {Zp pnq, Z, and Gm . The analogue of FPB(s,n) for morphisms between smooth varieties over fields are known (cf. [18] property 4 following Thm 1.1). Theorem 2.11. Suppose π : X Ñ S is of relative dimension d “ 1 and has a section s : S Ñ X . If n ě 2 assume that s satisfies the condition FPB(s,n). Then, for any integer n, the complexes ZpnqS and Zpn ´ 1qS r´2s split off as direct summands of Rπ˚ ZpnqX . When writing R1 π˚ ZpnqX r´1s for the remaining summand we arrive at the canonical p decomposition Rπ˚ ZpnqX » ZpnqS ‘ R1 π˚ ZpnqX r´1s ‘ Zpn ´ 1qS r´2s. p Proof. Let n ě 1. As before, we will prove the theorem by exhibiting maps ϕ0 ψ0 ϕ2 ψ2 ZpnqS ÝÑ Rπ˚ ZpnqX ÝÑ ZpnqS , Zpn ´ 1qS r´2s ÝÑ Rπ˚ ZpnqX ÝÑ Zpn ´ 1qS r´2s that compose as ˜ ψ0 ψ2 ¸ ˜ ´ ˝ ¯ ϕ0 ϕ2 “ id 0 0 id ¸ . (2.14) 31 Let Φ0 : π ˚ ZpnqS Ñ ZpnqX be the flat pull-back morphism, i.e. the adjoint of the canonical morphism ZpnqS Ñ π˚ ZpnqX which, on the level of complexes over an étale neighborhood U Ñ S, is given by Z n pU, jq ÝÑ Z n pπ ´1 U, jq, Z ÞÑ pπ ˆ id∆j q´1 Z. Similarly, let Ψ0 : s˚ ZpnqX Ñ ZpnqS denote the pull-back morphism (2.13). By virtue of the usual adjunctions, we may now define Rπ˚ Φ0 ϕ0 : ZpnqS ÝÑ Rπ˚ π ˚ ZpnqS ÝÝÝÝÑ Rπ˚ ZpnqX , 0 ψ0 : Rπ˚ ZpnqX ÝÑ Rπ˚ s˚ s˚ ZpnqX “ s˚ ZpnqX ÝÑ ZpnqS . Ψ One sees directly that ψ0 ϕ0 “ id, i.e. ZpnqS splits off as a direct summand of Rπ˚ ZpnqX . Next, we let Φ2 : s˚ Zpn ´ 1qS r´2s Ñ ZpnqX to be the morphism which acts on complexes as Z n´1 ps´1 U, jq Ñ Z n pU, jq, Z ÞÑ ps ˆ id∆j qpZq. Cor. 3.2 in [12] shows that Φ2 is well-defined. Note that the adjoint of Φ2 is a quasiisomorphism Zpn ´ 1qS r´2s » Rs! ZpnqX showing cohomological purity for Bloch’s cycle complexes (cf. [12] Cor. 7.2(a), Cor. 3.3(a)). Rπ˚ Φ2 ϕ2 : Zpn ´ 1qS r´2s “ Rπ˚ s˚ Zpn ´ 1qS r´2s ÝÝÝÝÑ Rπ˚ ZpnqX . Finally, let Ψ2 : π˚ ZpnqX Ñ Zpn ´ 1qS r´2s be the proper push-forward map which is given on cycles by n Z pπ ´1 U, jq Ñ Z n´1 pU, jq, Z ÞÑ $ & pπ ˆ id j qpZq ∆ % 0 if for all closed points x P ∆j : ZXpX ˆtxuq has codimension 2 in X ˆtxu, or is a finite union of horizontal divisors of X ˆtxu otherwise (2.15) The conditions on S guarantee that [12] Cor. 3.2 and Cor. 7.2(b) are applicable, showing that Ψ2 extends to a morphism ψ2 : Rπ˚ ZpnqX ÝÑ Zpn ´ 1qS r´2s. Again it is clear that ψ2 ϕ2 “ id. Also, from the explicit descriptions (2.13) and (2.15) we see immediately that the compositions ψ2 ϕ0 and ψ0 ϕ2 must be trivial. Consequently, ZpnqS and Zpn ´ 1qS r´2s split off as distinct direct summands of Rπ˚ ZpnqX . Let us now consider Zp´nq. Recall that Zp´nqX r1s “ à p jp,! Qp {Zp p´nq. 32 It suffices to show that Rπ˚ à jp,! “ à 1 jp,! Rπr1{ps˚ p (2.16) p 1 Q {Z p´nq. However, (2.16) is as the claim then follows from applying Theorem 2.8 to jp,! p p immediate from diagram (2.8) since π˚ “ π! and πr1{ps˚ “ πr1{ps! . Finally, for n “ 0 the Remark 2.9 shows Rπ˚ Z » Z ‘ τ ě1 Rπ˚ Z. Moreover the long exact sequence for the derived functor Rπ˚ associated to 0 ÝÑ ZX ÝÑ QX ÝÑ Q{ZX ÝÑ 0 proves R1 π˚ Z “ 0 and τ ě2 Rπ˚ Z “ τ ě1 Rπ˚ Q{Zr´1s. Indeed, one has Rr π˚ Q “ 0 for r ą 0 as can be seen by passing to stalks and recalling that Galois cohomology with rational coefficients vanishes. We may thus write Rπ˚ ZX “ ZS ‘ à τ ě1 Rπ˚ Qp {ZX p r´1s. (2.17) p For each p the Open-Closed-Decomposition (2.8) gives rise to the short exact sequence X 0 ÝÑ pQp {Zp qX ,p ÝÑ Qp {ZX ÝÑ ip,˚ Qp {Zp p ÝÑ 0. p Applying Rπ˚ yields the distinguished triangle X Rπ˚ pQp {Zp qX ÝÑ Rπ˚ Qp {ZX ÝÑ i1p,˚ Rπp,˚ Qp {Zp p ÝÑ . p p (2.18) X Rπp,˚ Qp {Zp p is concentrated in degrees 0, 1 and an analysis of the long exact sequence associated to (2.18) shows that applying τ ě1 preserves exactness. This yields ´ ¯ Xp 1 1 ě1 X ,p τ ě1 Rπ˚ Qp {ZX » Cone i R π Q {Z r´2s ÝÑ τ Rπ pQ {Z q . p p,˚ p ˚ p p p p,˚ We apply Theorem 2.8 to τ ě1 Rπ˚ pQp {Zp qX ,p and verify on stalks that there are no nonX trivial morphisms of sheaves i1p,˚ R1 πp,˚ Qp {Zp p Ñ pQp {Zp qS,p p´1q. Thus, we may rewrite the above as ´ ¯ p 1 Xp 1 1 X ,p τ ě1 Rπ˚ Qp {ZX r´1s ‘ pQp {Zp qS,p p´1qr´2s. p » Cone ip,˚ R πp,˚ Qp {Zp r´2s ÝÑ R π˚ pQp {Zp q Combining this with (2.17) and making the identification ´ à p 1 X R π˚ ZX :“ Cone i1p,˚ R1 πp,˚ Qp {Zp p r´2s ÝÑ R π˚ pQp {Zp qX ,p r´1s p 1 p allows us to write Rπ˚ ZX » ZS ‘ R1 π˚ ZX r´1s ‘ Zp´1qS r´2s p as desired. ¯ 33 Remark 2.12. The proof of Theorem 2.11 uses the regularity assumption as follows. For n ď 0 it is implicit in the use of Theorem 2.8 where in turn it is needed for the version of Verdier Duality given in Corollary 2.7. For n ě 1 it is implicit in the use of FPB(s,n) since this conjecture is formulated only for regular X . We believe FPB(s,n) to hold only for regular X . The most important instance of Theorem 2.11 is for the structure map π : X Ñ S of our arithmetic surface X as it will allow us to decompose motivic and Weil-étale motivic cohomology into degree 0, 1, 2 components. However, it is also applicable to localizations πZp : XZp Ñ SZp as well as to structure maps of smooth proper curves over fields. Note that for an elliptic surface π : E Ñ S one always has a section s : S Ñ E. In fact, EF has a rational point and EpOq “ EF pF q since E is proper (cf. [31] Cor. IV.4.4(a)). In the case of a general arithmetic surface π : X Ñ S one still has an exact triangle for n “ 1 without assuming the existence of a section. Let PX {S and PX0 {S denote the étale sheafifications of the functors U{S Ñ Pic pX ˆS Uq and U{S Ñ Pic0 pX ˆS Uq on S respectively. Proposition 2.13. One has the distinguished triangle Zp1qS ÝÑ Rπ˚ Zp1qX ÝÑ PX {S r´2s ÝÑ . X S Proof. Clearly, τ ď1 Rπ˚ Zp1qX “ pτ ď0 Rπ˚ GX m qr´1s “ π˚ Gm r´1s “ Gm r´1s giving us the truncation triangle Zp1qS ÝÑ Rπ˚ Zp1qX ÝÑ pτ ě1 Rπ˚ GX m qr´1s ÝÑ . Moreover, it is well-known that R1 π˚ GX m – PX {S . (2.19) It remains to show Ri π˚ GX m “ 0 for i ě 2. Let x ãÑ X be a geometric point over p. Write sh and let π be the base change of π to S . Then Sp “ Spec Opur and X pxq “ X ˆS OX x p ,x ` i ˘ i R π˚ GX m x “ H pX pxq, Gm q “ 0 for i ě 2 by Grothendieck’s result [14] Cor. 3.2 (p.98) applied to the surface X pxq as it is proper and flat over the spectrum Sp of a regular local ring. p Corollary 2.14. If π : X Ñ S has a section then R1 π˚ Zp1q – PX0 {S r´1s and one has the motivic decomposition Rπ˚ Zp1qX » Zp1qS ‘ PX0 {S r´1s ‘ ZS r´2s. 34 2.2.4 Motivic decompositions for complex manifolds. The analogue of Verdier duality and of Proposition 2.5 for locally compact spaces is well-known for all abelian sheaves. We use it to derive an analogue of Theorem 2.11 for the cohomology of complex manifolds with coefficients given by the locally constant GR -equivariant sheaf Rpnq :“ p2πiqn R. Proposition 2.15. Let S “ t‚u be the one-point space and let π : X Ñ S be the structure map of a complex manifold X of complex dimension d. Then the sheaves RpnqS and Rpn´dqS r´2ds p split off as direct summands of Rπ˚ RpnqX . If d “ 1 we will write R1 π˚ RpnqX r´1s for the remaining summand, i.e. we will have the canonical decomposition Rπ˚ RpnqX » RpnqS p ‘ R1 π˚ RpnqX r´1s ‘ Rpn ´ 1qS r´2s » R0 π˚ RpnqX ‘ R1 π˚ RpnqX r´1s ‘ R2 π˚ RpnqX r´2s. (2.20) Proof. Write p´q_ “ R Hom X p´, Rq. Smoothness of X implies π ! RX » RpdqS r2ds. Thus, the analogue of Verdier duality yields Rπ˚ RpdqX r2ds » Rπ˚ R Hom X pRX , RpdqS r2dsq » R Hom S pRπ˚ RX , RS q “ pRπ˚ RX q_ . Now, the proof of Theorem 2.8 holds verbatim for the structure map π : X Ñ S of complex manifolds with their analytic topology when replacing (2.9) with the above duality. This yields the first line of (2.20). Equality of motivic and cohomological degree components follows analogously to Remark 2.10. 2.2.5 The motivic picture and notation Motivic Interpretation. The decompositions of the previous propositions are motivated by the theory of motives over a field K. We recall it here. Any smooth projective variety X over K comes with an associated motive hpXq, an object in a Q-linear semi-simple abelian category MotQ . Conjecturally, hpXq produces H ˚ pXq for any Weil cohomology theory H by applying an appropriate fiber functor. One of the Standard Conjectures postulates the existence of algebraic cycles π i Ă X ˆ X that induce the projections H ˚ pXq  H i pXq onto ř i d the i-th degree. It would follow that ∆X “ 2d i“0 π in C„ pX ˆ Xq where d “ dim X. On the level of motives we would obtain the decomposition hpXq “ h 0 pXq ‘ ¨ ¨ ¨ ‘ h 2d pXq (2.21) of hpXq into its motivic degree components h i pXq. If X is a smooth projective curve over K the existence of a section, i.e. a K-rational point x P XpKq yields the above decomposition — even in the Z-linear category of Chow 35 motives — by setting π 0 “ X ˆ txu and π 2 “ txu ˆ X. Indeed, on cohomology the sequence txu ãÑ X  txu splits off H ˚ ptxuq “ H 0 pXq and taking the transpose cycle π 2 “ pπ 0 qt corresponds to projecting onto the Poincaré dual H 0 pXq_ – H 2 pXq. The remaining direct summand must be h 1 pXq and we may rewrite (2.21) as hpXq “ hpxq ‘ h 1 pXq ‘ hpxq_ p´1q. It is expected that a similar theory holds for proper regular X of relative dimension 1 over more general base schemes S. It should provide something analogous to the perverse t-structure on the derived category of l-adic sheaves on varieties over finite fields. A section s : S Ñ X would then give rise to an analogous decomposition p p p hpX q “ hpSq ‘ h 1 pX q ‘ hpSq_ p´1q. Applying the appropriate fiber functors should then reproduce (2.10) and (2.14). Notation. Let A P DpX q be a complex in the derived category of sheaves on any fixed À p topology of X . Whenever a decomposition of the kind Rπ˚ A “ i“0,1,2 Ri π˚ Ar´is holds p we will call Ri π˚ Ar´is the motivic degree i component or just shortly h i -component of Rπ˚ A, and we will write p p H i pX , Aq :“ H i pX , Ri π˚ Ar´isq. For example, Theorem 2.11 implies H i pX , Zpnqq “ à p H i pX , Zpnqq p“0,1,2 “ H i pS, Zpnqq ‘ 1H i pX , Zpnqq ‘ H i´2 pS, Zpn ´ 1qq. 2.3 Deligne Cohomology For the remainder of this thesis we assume that π : X Ñ S has a section s : S Ñ X satisfying the functorial pull-back condition FPB(s,n) for all integers n. Let X be a complex manifold and n ě 0. Recall that for a subring A Ă C one defines ApnqD “ ApnqX D as the bounded complex in the derived category of abelian sheaves on X given by 0 Ñ p2πiqn A Ñ OX{C Ñ ΩX{C Ñ . . . Ñ Ωn´1 X{C Ñ 0 concentrated in degrees r0, ns. One defines i HD pX, Apnqq :“ Hi pX, ApnqD q. 36 i pX pCq, Rpnqq For any arithmetic scheme X , the action of GR on X pCq carries through to HD and we define Deligne cohomology to be i,n i i HD pX q :“ HD pX {R , Rpnqq :“ HD pX pCq, RpnqqGR (cf. [30] §2 or [7] §1). The set of C-points X pCq of our arithmetic surface X has complex dimension 1. So, we have ΩiX pCq{C “ 0 for i ě 2 and Poincaré’s Lemma proves C » rOX pCq Ñ ΩX pCq{C s. Consequently $ ’ for n ď 0 ’ & “Rpnqr0s ‰ X pCq ˆ 1 RpnqD » Rp1q Ñ OX pCq » OX pCq {S r´1s for n “ 1 ’ ’ % rRpnq Ñ Cs » Rpn ´ 1qr´1s for n ě 2. (2.22) Here the pseudo-isomorphism for n “ 1 is given by the exponential map. Using (2.22) we reduce to singular cohomology and considering real and complex places separately gives us the i,1 table of ranks (A.12). For the computation of HD pX q with i ě 2 the perfect pairing 3´i i 3 HD pX {R , Rpnqq ˆ HD pX {R , Rp2 ´ nqq Ñ HD pX {R , Rp2qq Ñ R (2.23) from [8] Lemma 2.3 has been used. Decomposition into h i -components. It is easy to see that # RpnqSpCq for n ď 0 SpCq RpnqD » SpCq Rpn ´ 1q r´1s for n ě 1. Therefore, Proposition 2.15 shows together with (2.22) that also the Deligne complex decomposes as X pCq Rπ˚ RpnqD SpCq » RpnqD X pCq p ‘ R1 π˚ RpnqD SpCq r´1s ‘ Rpn ´ 1qD r´2s. On cohomology we obtain ‚,n ‚,n ‚,n ‚´2,n´1 HD pX q – HD pSq ‘ 1HD pX q ‘ HD pSq (2.24) i,n i,n and the motivic degree 1 term 1HD pX q equals the full HD pX q if i “ 1, n ď 0 or i “ 2, n ě 2, and vanishes otherwise. 2.4 Étale motivic cohomology For finitely or cofinitely generated groups we will write G „ H if G, H are isomorphic up to 2-torsion1 . We write Gdiv and Gcodiv “ G{Gdiv for the divisible and codivisible part 1 α β i.e. there are homomorphisms G Ñ H and H Ñ G such that kernel and cokernel of the compositions αβ and βα are finite 2-torsion groups 37 of G. We also let p´q_ “ Homp´, Q{Zq as well as p´q˚ “ Homp´, Zq for abelian groups and p´q˚ “ Homp´, Rq for R-vectorspaces respectively. For the remainder of this thesis we assume the validity of the following Conjecture 2.16 (L(X ,n)). The groups H i pX , Zpnqq are finitely generated for i ď 2n ` 1 and vanish for sufficiently small i. L(X ,1) is equivalent to finiteness of Br X (cf. [8] Lemma 3.3 and preceding comments). Assuming L(X ,n) allows us to reformulate Artin-Verdier duality as the existence of a perfect pairing of integral motivic cohomology groups (cf. [8] Prop. 3.4) H 6´i,2´n pX q ˆ H i,n pX q ÝÑ Q{Z. 2.4.1 (2.25) Completed motivic cohomology Recall the Artin-Verdier étale topos X ét and its open closed decomposition (1.1). We write H i,n pX q :“ H i pXét , Zpnqq and H i,n pX q :“ H i pX ét , Zpnqq for the motivic cohomology and for its completed cohomology, i.e. its cohomology with respect to the Artin-Verdier étale topos of X . The discrepancy between these two versions of cohomology is captured by the distinguished triangle ZpnqX ÝÑ Rφ˚ Zpnq ÝÑ u8,˚ τ ąn Rp π˚ p2πiqn Z ÝÑ . (2.26) (cf. [8] Cor. 6.8) In particular, for i ď n one has H i,n pX q “ H i,n pX q and for i ą n these cohomology groups differ only in 2-torsion. The cohomology of u8,˚ τ ąn Rp π˚ p2πiqn Z is computed in Appendix A.2. In what follows we will primarily work with completed motivic cohomology H i,n pX q as it is this type of cohomology that will factor into the definition of Weil-étale cohomology (see Section 2.4.5). As shown in Corollary A.12 in Appendix A.3 completed motivic cohomology also comes with a decomposition into motivic degrees H i,n pX q – H i,n pSq ‘ 1H i,n pX q ‘ H i´2,n´1 pSq. We will explicate this decomposition for n “ 1. 38 The case n “ 1 2.4.2 Proposition 2.17. The groups H i,1 pX q are given by and decompose as in the table below. i“1 H i,1 pX q Oˆ H i,1 pSq Oˆ 1 H i,1 pX q i“3 i“4 i“5 iě6 Pic pX q XpX{F q Pic pX q_ pOˆ q_ 0 ClF 0 0 0 0 0 pOˆ q_ 0 0 0 H i´2,0 pSq i“2 Pic pX q{ClF 0 Q{Z XpX{F q ` 0 Pic pX q{ClF 0 Z ˘_ ClF Proof. We use Lemma A.8(ii) to evaluate the long exact sequence on cohomology associated to the version of (2.26) for the base scheme S. In degrees 3, 4 we obtain b 0 Ñ H 3,1 pSq Ñ Br O ÝÑ pZ{2qr Ñ Q{Z Ñ H 4,1 pSq Ñ 0. ř Since Br O “ pZ{2qr, “0 the map b must be the inclusion and we get H 3,1 pSq “ 0 and H 4,1 pSq “ Q{ 12 Z – Q{Z as well as H 1,0 pSq “ H 1,0 pSq “ 0. Therefore the motivic decomposition of H 3,1 pX q “ Br X is given by Br X – Br O ‘ Br X . Br O So, the motivic degree 1 part of the triangle (2.26) gives 0 ÝÑ 1 H 3,1 pX q ÝÑ Br X 1 ÝÑ pZ{2q lpX q . Br O By [35] Thm. 3.1 and (1.7) the Tate-Shafarevich group XpX{F q fits into the exact sequence 0 ÝÑ XpX{F q ÝÑ ź Br X ÝÑ H 1 pGR , JacXσ q, Br O σ where σ runs through all finite places of F and Xσ “ X ˆF,σ Spec C. Proposition A.14 shows that the right-most terms of the above sequences are the same, so we in fact have 1 H 3,1 pX q – XpX{F q. Up to 2-torsion the remaining entries are immediate from Zp1q » Gm r´1s and Zp0q » Z or follow from Artin-Verdier duality. The additional 2-torsion information is taken from Proposition A.15. 2.4.3 Compact support cohomology and the perfect pairing conjecture Fan constructs in his thesis a map between complexes ρ : RΓpXét , Zpnqq Ñ RΓD pX {R , Zpnqq in the derived category of abelian groups that induces the Beilinson regulator maps H 2n´i,n pX q – 39 2n´i,n CHn pX , iq Ñ HD pX q. We define RΓc pX , Rpnqq as the mapping fiber of ρ b R, i.e. we have the distinguished triangle ρbR RΓc pX , Rpnqq ÝÑ RΓpXét , Rpnqq ÝÝÝÝÑ RΓD pX {R , Rpnqq ÝÑ (2.27) and write Hci,n pX q :“ Hci pX , Rpnqq for its cohomology groups. We have seen earlier that the motivic degrees of RΓD pX {R , Rpnqq coincide with cohomological degrees. So, ρ b R trivially decomposes into maps between the motivic degree components of RΓc pX , Rpnqq and RΓpXét , Rpnqq and we get a motivic decomposition of Hci,n pX q as well: Hci,n pX q “ Hci,n pSq ‘ 1Hci,n pX q ‘ Hci´2,n´1 pSq. Flach and Morin have constructed a product map RΓpX , Rpnqq b RΓc pX , Rpmqq ÝÑ RΓc pX , Rpn ` mqq (2.28) (cf. [8] Prop. 2.1) and have shown that under certain assumptions (cf. [8] Conj 2.9) Beilinson’s conjecture (cf. [30] §3) is equivalent to Conjecture 2.18. B(X ,n) The product map (2.28) induces for all i, n P Z a perfect pairing of R-vectorspaces Hci,n pX q ˆ H 4´i,2´n pX qR ÝÑ Hc4,2 pX q Ñ R. (2.29) Remark 2.19. B(X ,n) is equivalent to non-degeneracy of the induced pairing 1 Hci,n pX q ˆ 1H 4´i,2´n pX qR ÝÑ R. (2.30) In fact, due to the decompositions (2.14) and (2.10) the conjecture B(X ,n) is implied by the above together with B(S,n) and B(S,n ´ 1). However, B(S,n) is known for all n. We will later see that (2.30) for n “ 1 coincides with the height pairing which is known to be non-degenerate. In particular, B(X ,1) is a well-known fact. Ranks of motivic cohomology groups. From now on we assume B(X ,n) to hold for all n P Z. We use it to compute H i,n pX qR and Hci,n pX q. Together with cofinite generation of the H i,n pX qR for i ą 2n one gets # H i,n pX qR – R for i “ n “ 0 0 for n ă 0 or n “ 0, i ‰ 0 . So, we can read off the ranks of Hci,n pX q for n ě 1 from B(X ,n). Moreover, the long exact sequences associated to (2.27) for n ‰ 1 give us # 0,0 HD pX q{H 0,0 pX qR for i “ 1, n “ 0 i,n Hc pX q – i´1,n HD pX q for n ă 0 and i ‰ 1, n “ 0 40 and # H i,n pX qR – 3,2 KerpHD pX q Ñ Hc4,2 pX qq for i “ 3, n “ 2 3´i,2´n i,n HD pX q˚ – HD pX q for n ą 2 or i ‰ 3, n “ 2 . We obtain ranks as given in tables (A.15) and (A.17) in Appendix A.5. In particular, $ 1 1,n mg ’ if i “ 2, n ď 0 ’ & HD pX q – R 1 i,n 1 2,1 0 Hc pX q “ H pX qR “ pPic X {ClF qR if i “ 2, n “ 1 . ’ ’ % 0 otherwise 2.4.4 Torsion of motivic cohomology Let T?i,n “ Tor H i,n p?qcodiv for ? “ S, S, X , X as well as 1T?i,n “ Tor 1H i,n p?qcodiv for ? “ X , X . Artin-Verdier duality gives TSi,n – TS4´i,1´n . Moreover, it is known that for n ě 2, i ‰ 1, 2 one has TSi,n „ 0 and even TSi,n “ 0 for i ď 0 (cf. [8] Section 5.8.3). In this section, we establish an analogous vanishing result for the torsion parts of the h 1 -part of the motivic cohomology of X . Proposition 2.20. Let n be any integer. One has 1TXi,n „ 1TXi,n „ 0 whenever i ‰ 2, 3, 4 and, moreover, 1TXi,n “ 0 for i ă 2. Proof. Due to Artin-Verdier duality it suffices to consider i ă 2. For n ă 0 the claim is À immediate from the definition Zpnq “ p jp,! Qp {Zp p´nq and for n “ 0, 1 it follows from the explicit expressions Zp0q » Z and Zp1q » Gm r´1s. Let now n ě 2 and fix a prime p. We will show that 1H i pX , Zp pnqq “ 0 for i ď 0, proving that 1TXi,n has trivial p-part for i ď 1. Consider the Open-Closed-Decomposition j i X r1{ps ÝÑ X ÐÝ XFp . The proof of [8] Lemma 7.7 provides the distinguished triangle i˚ Ri! Zp pnq ÝÑ Zp pnq ÝÑ Rj˚ Zp pnq ÝÑ (2.31) together with a quasi-isomorphism » τ ďn`1 pi˚ Zp pn ´ 1qr´2sq ÝÑ τ ďn`1 i˚ Ri! Zp pnq. Zp pn ´ 1qXFp is known to be cohomologically concentrated in degrees rpn ´ 1q, 2pn ´ 1qs (see [38] Thm. 1.1). Consequently, τ ďn RΓpi˚ Ri! Zp pnqq » τ ďn RΓpτ ďn i˚ Ri! Zp pnqq ` ˘ » τ ďn RΓ τ ďn pi˚ Zp pn ´ 1qr´2sq » 0. 41 Therefore (2.31) implies τ ăn RΓpX , Zp pnqq » τ ăn RΓpX , Rj˚ Zp pnqq » τ ăn RΓpX , Rj˚ τ ďn Zp pnqq ` “ ‰ ˘ ăn » τ ăn RΓpX , Rj˚ µbn RΓ X p1 , µbn p8 q » τ p8 , (2.32) where the third quasi-isomorphism is [38] Thm. 2.6. Theorem 2.8 provides the decomposition p bpn´1q bn bn 1 Rπr1{ps˚ µbn p‚ » µp‚ ‘ R πr1{ps˚ µp‚ r´1s ‘ µp‚ r´2s. A direct comparison of the cohomology groups of both sides (or, alternatively, Remark 2.9) p shows that R1 πr1{ps˚ µbn p‚ r´1s is concentrated in positive degrees. So, by virtue of (2.32), one has p p τ ă1 R1 ΓpX , Zpnq{p‚ q » τ ă1 R1 ΓpX r1{ps, µbn p‚ q ` ˘ p » τ ă1 RΓ X r1{ps, R1 πr1{ps˚ µbn p‚ r´1s “ 0, proving the proposition. 2.4.5 Weil-étale cohomology Flach’s and Morin’s work in [8] is founded on their insight that even in the absence of any Weil-étale topos one may construct a Weil-étale cohomology complex RΓW pX , Zpnqq — in terms of which the special value conjectures are then formulated — utilizing Artin-Verdier duality. We recall their definitions. Flach and Morin use perfectness of the pairing (2.25) to construct a morphism αX ,n : RHompRΓpX , Qp2 ´ nqq, Qr´6sq ÝÑ RΓpX , Zpnqq (cf. [8] Thm. 3.5) whose induced maps on cohomology H i pαX ,n q have image equal to the divisible part of H i,n pX q. In other words, they factor as follows: H i pαX ,n q : HomQ pH 6´i,2´n pX q b Q, Qq  H i,n pX qdiv ãÑ H i,n pX q. (2.33) Weil-étale cohomology RΓW pX , Zpnqq is defined as the mapping cone of αX ,n , i.e. one has a distinguished triangle RHompRΓpX , Qp2 ´ nqq, Qr´6sq αX ,n / RΓpX , Zpnqq / RΓW pX , Zpnqq /. i,n We write HW pX q :“ H i pRΓW pX , Zpnqqq. From the associated long exact sequence and the factoring (2.33) one easily deduces i,n HW pX q – H i,n pX qcodiv ‘ HompH 5´i,2´n pX q, Zq. (2.34) 42 i,n In particular, L(X ,n) ensures that all HW pX q are finitely generated. (2.34) gives the Weil-étale cohomology groups listed in (A.18) and for n “ 1 we get the splittings i,1 HW pX q i,1 HW pSq 1 i,1 HW pX q i´2,0 HW pSq 2.5 i“1 i“2 Oˆ Pic X Oˆ ClF 0 0 0 i“3 i“4 i“5 iě6 µF 0 XpX{F q ‘ pPic X q˚ Tor Pic X ‘ pOˆ q˚ 0 0 0 Pic X {ClF XpX{F q ‘ pPic X q˚ 0 Tor Pic X { ClF 0 0 Z 0 ClF ‘ pOˆ q˚ µF 0 Z 0 Betti cohomology and Weil-étale cohomology with compact support The long exact sequence induced by the regulator map splits motivic cohomology into a compactly supported part and an infinite part given by Deligne cohomology. In this section we work out the analogous decomposition for Weil-étale motivic cohomology. Betti cohomology. Let RΓW pX8 , Zpnqq be defined via the exact triangle RΓW pX8 , Zpnqq ÝÑ RΓpGR , X pCq, p2πiqn Zq ÝÑ RΓpX pRq, τ ąn Rp π˚ p2πiqn Zq ÝÑ (2.35) (cf. [8] Def. 3.23). Since the rightmost complex is entirely 2-torsion we have RΓW pX8 , Zpnqq b R » RΓpGR , X pCq, p2πiqn Zq b R » RΓpX pCq, RpnqqGR . i,n We write HW,8 pX q :“ H i pRΓW pX8 , Zpnqqq. We evaluate the singular cohomology groups on i,n the right hand side directly and get ranks as in table (A.13). The torsion groups of HW,8 pX q are computed in Appendix A.2. Lemma A.11 shows that RΓpX pRq, τ ąn Rp π˚ p2πiqn Zq and RΓpGR , X pCq, p2πiqn Zq decompose into motivic degrees. Consequently, the entire triangle (2.35) decomposes and we have i,n i,n i,n i´2,n´1 HW,8 pX q – HW,8 pSq ‘ 1HW,8 pX q ‘ HW,8 pSq. Compactly supported Weil-étale cohomology. There is a canonical map u˚8 : RΓpX ét , Zpnqq Ñ RΓW pX8 , Zpnqq since RΓpX ét , Zpnqq can be regarded as the mapping fiber of the composition π˚ p2πiqn Zq. RΓpXét , Zpnqq Ñ RΓD pX {R , Zpnqq Ñ RΓpGR , X pCq, p2πiqn Zq Ñ RΓpX pRq, τ ąn Rp Flach and Morin have shown that there is a unique i˚8 : RΓW pX , Zpnqq Ñ RΓW pX8 , Zpnqq making u˚8 factor through RΓW pX , Zpnqq (cf. [8] Prop. 3.24). RΓW,c pX , Zpnqq is defined as 43 the mapping fiber of i˚8 , i.e. we have a short exact triangle i˚ 8 RΓW pX8 , Zpnqq ÝÑ . RΓW,c pX , Zpnqq ÝÑ RΓW pX , Zpnqq ÝÑ (2.36) i,n We write HW,c pX q :“ H i pRΓW,c pX , Zpnqqq for its cohomology groups. As u˚8 splits into h i -parts, so does i˚8 and we obtain the usual decomposition i,n i,n i,n i´2,n´1 HW,c pX q “ HW,c pSq ‘ 1HW,c pX q ‘ HW,c pSq. i,n We will evaluate HW,c pX q later, as part of the computation of fundamental lines. 2.6 De Rham and derived de Rham cohomology Algebraic de Rham cohomology. Let Π : X Ñ S be an arithmetic scheme. Recall algebraic de Rham cohomology RΓdR pX {Sq :“ RΓpX , Ω‚X {S q and its Hodge filtration given by RΓdR pX {Sq{Fn :“ RΓpX , Ω‚X {S {Filn q where n´1 ‚ Ωăn X {S :“ ΩX {S {Filn :“ rOX Ñ ΩX {S Ñ . . . Ñ ΩX {S s as a complex in the derived category of sheaves of OX -modules on XZar concentrated in degrees r0, n ´ 1s. The de Rham complexes of the generic fiber X over Q, of X8 over R, and of X pCq over C are related as follows: RΓdR pX8 {Rq{F n “ RΓdR pX{F q{F n b R “ pRΓdR pX pCq{Cq{F n qGR . Q We call RΓdR pX8 {Rq{F n the real de Rham complex and write i,n GR HdR pX q :“ H i pRΓdR pX8 {Rq{F n q “ Hi pX pCq, Ωăn X pCq{C q . For n ă 1 these groups are trivial. For n “ 1 de Rham cohomology simplifies to the i,1 cohomology of the structure sheaf HdR pX q “ H i pX pCq, OX pCq qGR which is well-understood. an For n ě 2 we use the GAGA principle to reduce to analytic cohomology. Recall that rOX pCq Ñ X pCq (cf. [37] Lem. 8.13) so that H i,2 “ H i pX pCq, CqGR . We Ωan an dR X pCq{C s is a resolution of C infer the ranks as given in table (A.14). We will now exhibit an integral structure for RΓdR pX{Qq{F n . The natural candidate arising from RΓdR pX {Zq{F n turns out to have undesirable properties for n ě 2. We thus resort to derived de Rham cohomology. 44 Derived de Rham cohomology. Derived de Rham cohomology is a variant of the above where the alternating powers ΩkX {S of the Kähler differentials are replaced with derived Ź alternating powers L k LX {S of the cotangent complex LX {S . Its construction and related notations can be found in Appendix A.4. We write i,n RΓddR pX {Sq{F n :“ RΓpXZar , LΩ‚X {S {F n q and HddR pX q :“ H i pRΓddR pX {Zq{F n q. Algebraic and derived de Rham cohomology coincide for smooth schemes. In particular – when denoting the generic fiber of X Ñ S by X Ñ Spec k – one has LΩ‚X {S b OX “ LΩ‚X {k » Ω‚X {k . Consequently, RΓddR pX {kq{F n “ RΓdR pX {kq{F n attains an integral structure via ´b1 RΓddR pX {Sq{F n ÝÝÝÝÑ RΓddR pX {Sq{F n b k “ RΓddR pX {kq{F n . For the remainder of this section we assume Π : X Ñ S to be projective and regular. In particular, X may be any proper regular arithmetic surface or curve. Let i : X Ñ PN S be a closed immersion into projective space and let I denote the sheaf of OPN -modules generated S by the equations defining X . Lemma 2.21. In the derived category of sheaves of OX -modules one has LX {S » ΩX {S . i Proof. The morphisms of schemes X ãÑ PN S Ñ S induce a distinguished triangle of cotangent complexes i˚ LPN {S ÝÑ LX {S ÝÑ LX {PN ÝÑ S S in the derived category DpModOX q of OX -modules (cf. [16] Prop. 2.1.2 and 2.1.5.6). As PN S Ñ S is smooth we have a quasi-isomorphism LPN {S » ΩPN {S (cf. [16] Ch. III, Prop. S S 3.1.2). Moreover, by [16], Prop.3.2.4(ii) one has LX {PN » I{I 2 r1s. So, we have S d I{I 2 ÝÑ i˚ ΩPN {S ÝÑ LX {S ÝÑ S in DpModOX q. Here d is given by the usual Kähler differential. d has to be injective as H ´1 pLX {S q “ 0. In fact, by [16] Prop. 3.2.6 one has LX {S “ rF Ñ Gsr`1s with F, G being finitely generated and locally free. Besides, one has LX {S b OX “ LX {k “ ΩX {k r0s. Therefore – as F is torsion-free – the map F Ñ G must be injective and, moreover, LX {S » H 0 pLX {S q. In particular, d LX {S “ rI{I 2 ÝÑ i˚ ΩPN {S sr1s » ΩX {S . S (2.37) The last quasi-isomorphism follows from direct inspection or, alternatively, from the short exact sequence (2.1) in [3]. 45 We will now compare algebraic and derived de Rham cohomology for π 1 : X Ñ Spec Z. We begin with some preparations. The second quasi-isomorphism in (2.37) can be rewritten as the short exact sequence 0 ÝÑ I{I 2 ÝÑ i˚ ΩPN {Z ÝÑ ΩX {Z ÝÑ 0. Z It provides a canonical map ρ1 which we regard as a two-term complex „  ρ1 1 2 ˚ C “ ΩX {Z b detOX I{I ÝÝÝÝÑ detOX i ΩPN {Z r1s Z in degrees ´1 and 0 or, equivalently, a map ρ giving rise to Bloch’s complex „  ρ 1 C “ ΩX {Z ÝÝÝÝÑ ωX {Z r1s, where ´ ¯ ωX {Z “ Hom detOX I{I 2 , detOX i˚ ΩPN {Z Z is the canonical normal bundle of X (cf. [3] §2 for Bloch’s treatment of ωX {Z , ρ, C ). For a complex C ‚ in the derived category of abelian groups we write ź` ˘p´1qi #Tor H i pC ‚ q χpC ‚ q :“ iPZ for its multiplicative Euler characteristic if it is well-defined. For a complex F ‚ of abelian sheaves on X we write2 χpF ‚ q :“ χpRΓpX , F ‚ qq :“ ź` ˘p´1qi #Hi pX , F ‚ q . iPZ Recall the definition of the conductor ApX q “ χpΩ‚X {Z,tors q. We will need Theorem 2.22. (Bloch, [3] Thm. 2.3) One has ApX q “ χpC q. Proposition 2.23. Let X be a regular arithmetic surface with proper structure map π 1 : X Ñ Spec Z. (i) One has gr0 pLΩX {Z q » OX and gr1 pLΩX {Z q » ΩX {Z r´1s. Moreover, for i ě 2 the graded piece gri pLΩX {Z q is locally quasi-isomorphic to C 1 r´2s, i.e. for any point x of X there is a quasi-isomorphism of complexes of stalks gri pLΩX {Z qx » Cx1 r´2s in the derived category of OX ,x -modules. In particular, for i ě 2 the complex gri LΩ‚X {Z is cohomologically concentrated in degrees 1 and 2. 2 This is an extension of Bloch’s notation in [3] which he only defines for complexes with torsion cohomology. This is, e.g., the case if the base change F ‚ b Q yields a bounded exact complex of finitely generated abelian sheaves. 46 (ii) Let n ě 2. Then F 2 {F n is cohomologically concentrated in degrees 1 and 2 and there is a distinguished triangle RΓpX , F 2 {F n q ÝÑ RΓddR pX {Zq{F n ÝÑ RΓpX , Ωď1 X {Z q ÝÑ . (2.38) (iii) Let n ě 2. We make the following technical assumption: RPpX q All special fibers of X are reduced, or there is a closed immersion i : X Ñ P2Z into two-dimensional projective space. Then one has detZ RΓddR pX {Zq{F n “ ApX qn´2 , detZ RΓddR pX {Zq{F 2 (2.39) where the left hand side is understood as a quotient of lattices in the (1-dimensional) Q-vectorspace detQ RΓdR pX{Qq{F n “ detQ RΓdR pX{Qq{F 2 . ď1 Proof. (i): LΩď1 X {Z » ΩX {Z is immediate from Lemma 2.21. Now, recall the identity LX {Z “ d rI{I 2 ÝÑ i˚ ΩPN {Z sr1s from its proof. Note that both OX -modules I{I 2 and i˚ ΩPN {Z are Z Z locally free of finite type. So, following Illusie we may express the derived exterior powers of LX {Z in terms of Koszul complexes. ´ ľi ¯ gri LΩX {Z » L LX {Z r´is [17] Ch. VIII, (2.1.1.5) (p.277) X » LΓX pLX {Z r´1sqi [16] I.4.3.2.1(ii), or proof of [17] VIII. Col. 2.1.2.2 d » Kosi pI{I 2 ÝÑ i˚ ΩPN {Z q [17] Ch. VIII Lem. 2.1.2.1 Z ” ı ľN i´N 2 ˚ 2 ˚ » ΓiOX I{I 2 Ñ Γi´1 I{I b i Ω Ñ . . . Ñ Γ I{I b i Ω N N P {Z P {Z . OX OX Z Z (2.40) Let i ě 2. We treat the case N “ 2 first. Then there is a section f generating I, and i˚ ΩPN {Z Z as well as Γi I{I 2 are locally free of rank 2 and rank 1 respectively. Write γj pf q for the generator of Γj I{I 2 . Then γ1 pf q “ f . Since I{I 2 is a line bundle one has Γj I{I 2 – pI{I 2 qbj . 47 Thus, the Koszul complex simplifies to i gr LΩX {Z rΓiI{I 2 » γi pf q ÝÑ Γi´1 I{I 2 b i˚ ΩP2 {Z Z Ź2 ˚ i ΩP2 {Z s Z γi´1 pf q b df ÞÑ γi´1 pf q b ω » Γi´2 I{I 2 b ÝÑ pI{I 2 qbpi´2q b γi´2 pf q b df ^ ω ÞÑ rpI{I 2qb2 Ñ I{I 2 b i˚ΩP {Z Ñ Ź2 i˚ΩP {Zs 2 Z f bf 2 Z f b df ÞÑ f bω » pI{I 2 qbpi´2q b r I{I 2 b Ω ´^ df f X {Z f bη ÝÝÝÝÑ df ^ ω ÞÑ Ź2 ˚ i ΩP2 {Z sr´1s Z η ^ df ÞÑ ρ1 » pI{I 2 qbpi´2q b rΩX {Z b detO I{I 2 ÝÝÝÝÑ detO i˚ΩP {Zsr´1s » pI{I 2 qbpi´2q C 1 r´2s. b X X 2 Z (2.41) Here we have used (2.37) and the fact that taking the tensor product with a line bundle is an exact operation. The claim now follows after passing to stalks. Since the claim is local the general case N ě 2 follows from the observation that X may locally be described by one equation f pu, vq “ 0 as a subscheme of Spec ZJu, vK (cf. [3], Proof of Lemma 2.4). (ii): We use the distinguished triangle F 2 {F n ÝÑ LΩ‚X {Z {F n ÝÑ LΩ‚X {Z {F 2 ÝÑ and apply RΓpX , ´q to obtain (2.38). Next, we consider for all m ě 2 grm LΩX {Z ÝÑ F 2 {F m`1 ÝÑ F 2 {F m ÝÑ (2.42) ď1 2 m are cohomologically concenand use LΩď1 X {Z » ΩX {Z to conclude inductively that all F {F trated in degrees 1 and 2. This completes (ii). (iii): Let i ě 2. First assume that X embeds into P2Z . (2.41) shows that there is a line bundle L on X such that gri LΩX {Z » L b C 1 r´2s. Therefore (2.42) gives χpF 2 {F i`1 q “ χpgri LΩX {Z q “ χpL b C 1 q “ χpL b C q. χpF 2 {F i q (2.43) 48 In fact, the third equality holds since ρ arises from ρ1 via the adjunction between b and Hom. We will show the identity χpL b C q “ χpC q (2.44) later in this proof. Using (2.44) and Theorem 2.22 one obtains χpF 2 {F n q “ ApX qn´2 inductively. Consequently – by virtue of (2.38) – detZ RΓddR pX {Zq{F n detZ RΓddR pX {Zq{F n “ ApX qn´2 . “ ď1 2 detZ RΓddR pX {Zq{F detZ RΓpX , ΩX {Z q (2.45) It remains to show (2.44). The distinguished triangle L b Kerpρqr1s ÝÑ L b C ÝÑ L b cokerpρq ÝÑ bcokerpρqq proves that χpL b C q “ χpL χpL bKerpρqq . The same can be said for χpC q. So, it suffices to prove χpL b cokerpρqq χpL b Kerpρqq “ . χpKerpρqq χpcokerpρqq (2.46) Kerpρq and cokerpρq are supported on the at most 1-dimensional subscheme Z Ă X of non-smooth points of X (since Kerpρq – ΩX {Z,tor and cokerpρq is locally isomorphic to Ω2X {Z ). Write Zp “ Z X Xp . We show (2.46) for the restrictions to each Zp separately. Fix a prime p. By abuse of notation we write L again for the restriction L |Zp . When writing χ̃ for the additive Euler characteristic3 then the Riemann-Roch Theorem for non-reduced curves (cf. [36] Ex. 18.4.S) gives for any coherent sheaf F on X ÿ χ̃pL b F q ´ χ̃pF q “ degZ red L |Z red ¨ lengthOη Fηi . i i i Zi ĂZp Here the sum is taken over all irreducible components Zi of Zp and ηi denotes the generic point of Zi . Now, (2.46) follows from applying the above formula to Kerpρq|Zp and cokerpρq|Zp and using Bloch’s result [3] Lemma 2.5 that lengthOη Kerpρqη “ lengthOη cokerpρqη for every codimension 1 point η of X . We now assume all special fibers of X to be reduced instead of having an embedding of X into P2Z . The subscheme Z Ă X of non-smooth points is then 0-dimensional. Write HBlj :“ H j pC 1 r´2sq and j HddR :“ H j pgri LΩX {Z q j for the cohomology sheaves in degrees j “ 1, 2. (2.41) shows that HBlj and HddR have isomorphic stalks. However, since the HBlj are supported on the finite collection of points Z 3 i.e. the alternating sum of the dimensions of the cohomology groups as kppq-vectorspaces. This means it bF q relates to χ via N pχ̃pL bF q´χ̃pF q “ χpL . χpF q 49 j this glues to global isomorphisms of sheaves HBlj – HddR for j “ 1, 2. From the pτ ď1 , τ ě2 q- truncation triangles for gri LΩX {Z and C 1 r´2s we obtain χpgri LΩX {Z q “ 2 q χpHddR χpHBl2 q “ “ χpC 1 q “ χpC q. 1 1 χpHddR q χpHBl q The claim follows as in (2.45). Remark 2.24. We expect (2.39) to also hold without the assumption RP(X ). However, it is unclear how to construct a line bundle L satisfying (2.43) from the local isomorphisms (2.41) without an embedding into P2Z . ‚ . Derived de Rham cohomology endows RΓ pX {Rq The canonical bundle complex ωX 8 dR {Z with integral structures for each n ě 2. It will turn out to be most natural to compare them to a further integral structure coming from the complex „  ρ˝d ‚ ωX {Z :“ OX ÝÝÝÝÑ ωX {Z . ‚ q “ RΓ pX{Qq. The advantages of using ω ‚ One indeed has RΓpX , ωX dR {Z Q X {Z are two-fold: ‚ q is torsion-free and its induced integral structure on (i) The cohomology of RΓpX , ωX {Z RΓdR pX{Qq will allow for a formula for the later to be defined correction factor CpX , nq that does not contain the conductor ApX q or any unspecified torsion cardinalities. ‚ q admits a motivic decomposition fitting neatly into the formalism developed (ii) RΓpX , ωX {Z in this chapter (cf. Proposition 2.30). Proposition 2.25. One has detZ RΓddR pX {Zq{F 2 “ ApX q. ‚ q detZ RΓpX , ωX {Z So, if RP(X ) holds, one has for any n ě 2 detZ RΓddR pX {Zq{F n “ ApX qn´1 . ‚ q detZ RΓpX , ωX {Z (2.47) Proof. Due to Proposition 2.23(iii) it suffices to prove the claim for n “ 2. In this case the left hand side equals detZ RΓpX , Ωď1 X {Z q ‚ q detZ RΓpX , ωX {Z ˆ “χ ρ Ω1X {Z ÝÝÝÑ ωX {Z r ˙ s “ ApX q, where the last equality is Theorem 2.22. Lemma 2.26. (cf. [3] Lemma 2.2) Write p´q˚ “ R Hom p´, Zq. Then one has ‚ ˚ ‚ pRπ˚ ωX {Z q » Rπ˚ ωX {Z r`2s. ‚ q is torsion-free. In particular, the cohomology of RΓpX , ωX {Z 50 Decomposition into motivic degrees. Proposition 2.27. Suppose πC : X pCq Ñ SpCq has a section sC : SpCq Ñ X pCq, i.e. for each real embedding σ of F one has Xσ pRq ‰ H. Let n P Z. Then RΓdR pS8 {Rq{F n and RΓdR pS8 {Rq{F n´1 r´2s split off as direct summands of RΓdR pX8 {Rq{F n . When writing p 1 R ΓdR pX8 {Rq{F n r´1s for the remaining summand we have a canonical decomposition p RΓdR pX8 {Rq{F n » RΓdR pS8 {Rq{F n ‘ R1 ΓdR pX8 {Rq{F n r´1s ‘ RΓdR pS8 {Rq{F n´1 r´2s in the derived category of abelian groups. Moreover, each summand on the right hand side is cohomologically concentrated in one degree only, i.e. 0,n 1,n 0,n´1 RΓdR pX8 {Rq{F n » HdR pS8 {Rqr0s ‘ 1HdR pX8 {Rqr´1s ‘ HdR pS8 {Rqr´2s. Proof. The claim is trivial for n ď 0. For n “ 1 it suffices to show that OSpCq splits off as a direct summand of RπC,˚ OX pCq . Since s˚C OX pCq “ OSpCq and πC˚ OSpCq “ OX pCq this follows verbatim as in the proof of Theorem 2.8. (Also cf. Remark 2.9). For n “ 2 the decomposition is immediate from Proposition 2.15. Finally, a comparison with the ranks in table (A.14) shows that each complex on the right hand side is concentrated in one degree only. Proposition 2.28. Suppose π : X Ñ S has a section s : S Ñ X . Then, for any integer n, the complex LΩ‚S{Z {F n splits off as a direct summand of Rπ˚ LΩ‚X {Z {F n . When writing p ě1 R π˚ LΩ‚X {Z {F n r´1s for the remaining summand we have a canonical decomposition p Rπ˚ LΩ‚X {Z {F n » LΩ‚S{Z {F n ‘ Rě1 π˚ LΩ‚X {Z {F n r´1s in the derived category of OF -modules. Proof. The cotangent complex formalism provides maps `π : π ˚ LΩ‚S{Z {F n Ñ LΩ‚X {Z {F n and `s : s˚ LΩ‚X {Z {F n Ñ LΩ‚S{Z {F n satisfying `s ˝ s˚ `π “ id. Consequently, the resulting maps Rπ˚ `π ϕ0 : LΩ‚S{Z {F n ÝÑ Rπ˚ π ˚ LΩ‚S{Z {F n ÝÝÝÑ Rπ˚ LΩ‚X {Z {F n , ` s ψ0 : Rπ˚ LΩ‚X {Z {F n ÝÑ Rπ˚ s˚ s˚ LΩ‚X {Z {F n “ s˚ LΩ‚X {Z {F n ÝÑ LΩ‚S{Z {F n compose to the identity on LΩ‚S{Z {F n . So, LΩ‚S{Z {F n splits off as a direct summand of Rπ˚ LΩ‚X {Z {F n proving the proposition. Remark 2.29. It is unclear whether to expect the existence of a full decomposition ? p Rπ˚ LΩ‚X {Z {F n » LΩ‚S{Z {F n ‘ R1 π˚ LΩ‚X {Z {F n r´1s ‘ LΩ‚S{Z {F n´1 r´2s (2.48) analogously to Theorem 2.11. In order to replicate its proof one would need a duality result of the kind pRπ˚ LΩ‚X {Z {F n q˚ » Rπ˚ LΩ‚X {Z {F n´1 r´2s (2.49) 51 for a suitable duality operation p´q˚ that is analogues to classical Verdier duality Theorem 2.3 or Geisser’s duality [12] Thm. 7.3. In any case, one has a decomposition of the integral structure determined by the canonical bundle complex. Proposition 2.30. Write ωF “ HompOF , Zq for the different ideal of OF . Suppose π : X Ñ ‚ . S has a section s : S Ñ X . Then OF and ωF r´2s split off as direct summands of Rπ˚ ωX {Z p ‚ r´1s for the remaining summand we have a canonical decomposition When writing R1 π˚ ωX {Z p ‚ 1 ‚ Rπ˚ ωX {Z » OF ‘ R π˚ ωX {Z r´1s ‘ ωF r´2s in the derived category of OF -modules. Proof. We mimic the proof of Theorem 2.8. Define ‚ ‚ ϕ0 : OF “ R0 π˚ ωX {Z ÝÑ Rπ˚ ωX {Z , ‚ ˚ ‚ ˚ ‚ ψ0 : Rπ˚ ωX {Z ÝÑ Rπ˚ s˚ s ωX {Z “ s ωX {Z ÝÑ OF . The composition ϕ0 ψ0 ‚ OF ÝÝÝÑ Rπ˚ ωX {Z ÝÝÝÑ OF (2.50) is the identity. Apply p´q˚ “ R Hom p´, Zq to (2.50) and then shift by ´2 degrees. Due to Lemma 2.26 one obtains ϕ˚ 0 r´2s ωF r´2s ÐÝÝÝÝÝÝÝ ‚ Rπ˚ ωX {Z ψ0˚ r´2s ÐÝÝÝÝÝÝÝ ωF r´2s. We let ϕ2 “ ψ0˚ r´2s and ψ2 “ ϕ˚0 r´2s. Again, one has ψ2 ϕ2 “ pϕ0 ψ0 q˚ r´2s “ id. Furthermore, ψ2 ϕ0 “ 0 and ψ0 ϕ2 “ 0 for degree reasons. In the absence of a duality result of type (2.49) we will introduce ad hoc definitions to artificially force a splitting of derived de Rham cohomology on the level of determinants. This allows us to use the formalism for motivic decompositions as developed in this chapter on derived de Rham cohomology as well. Definition 2.31. Let n be any integer. p p (i) Let R0 π˚ LΩ‚X {Z {F n :“ LΩ‚S{Z {F n and R2 π˚ LΩ‚X {Z {F n :“ LΩ‚S{Z {F n´1 r´2s. Write p p n´1 detZ R1 ΓddR pX {Zq{F n :“ detZ Rě1 ΓddR pX {Zq{F n b det´1 . Z RΓddR pS{Zq{F (ii) For i “ 0, 1, 2 let p detZ Ri ΓddR pX {Zq{F 2 . ApX q :“ p ‚ q detZ Ri ΓpX , ωX {Z i 52 (iii) For any integer n pnq tddR pX q :“ χpRΓddR pX {Zq{F n q pnq tddR pSq :“ χpRΓddR pS{Zq{F n q. and pnq Also, let 1tddR pX q be defined by the equation pnq pnq pnq pn´1q tddR pX q :“ tddR pSq ¨ 1tddR pX q´1 ¨ tddR pSq. Remark 2.32. p (i) The symbol R1 ΓddR pX {Zq{F n itself is undefined. However, Definition 2.31(i) ensures that we have the decompositions â ` detZ RΓddR pX {Zq{F n “ p detZ Ri ΓddR pX {Zq{F n ˘p´1qi . (2.51) i“0,1,2 as well as ź ApX q “ i i ApX qp´1q . (2.52) i“0,1,2 pnq Similarly, 1tddR pX q should be thought of as a substitute for the hypothetical Euler p characteristic χp R1 ΓddR pX {Zq{F n q. (ii) For any n ě 1 one has tnddR pSq “ ApSqn´1 “ p#DF qn´1 by [8] Prop. 5.35. Moreover, p1q tddR pX q “ 1 since H 1 pX , OX q – H 0 pX , ωX {Z q has no torsion and H 2 pX , OX q – H ´1 pX , ωX {Z q “ 0 by Serre duality. (iii) The exact triangle OX Ñ LΩ‚X {Z {F 2 Ñ ΩX {Z r´1s Ñ and the short exact sequence 0 Ñ ωF Ñ OF Ñ ΩS{Z Ñ 0 prove that 0 ApX q “ 2 ApX q “ detZ RΓddR pS{Zq{F 2 “ χpΩS{Z q “ ApSq, detZ RΓpS, OF q detZ RΓddR pS{Zq{F 1 “ χpΩS{Z q “ ApSq. detZ RΓpS, ωF q Therefore, the decomposition (2.52) becomes ApX q “ ApSq ¨ 1ApX q´1 ¨ ApSq. 2.7 Completions of L- and ζ-functions Let H i pX pCq, Cq “ p,q be the Hodge Decomposition and write hp,q “ dim H p,q . C p`q“i H p,p ˘1 Further, write pH q for the eigenspace of complex conjugation to the eigenvalue ˘1 and À p let hp,˘ “ dimC pH p,p q˘p´1q for integral p and hp,˘ “ 0 otherwise. Define ź ź p,q p,` p,´ L8 pH i pXq, sq :“ ΓC ps ´ pqh ¨ ΓR ps ´ pqh ΓR ps ´ p ` 1qh . p`q“i, păq p“q“ 2i 53 Obviously L8 pH i pXq, sq decomposes into a product over all infinite places of F similarly to (2.1). We analogously write ζpX8 , sq :“ ź i L8 pH i pXq, sqp´1q (2.53) iPZ and define the completions of ζpX , ´q and ζHW pX, ´q to be ζpX , sq :“ ζpX8 , sqζpX , sq, and ζHW pX, sq :“ ζpX8 , sqζHW pX, sq. Bloch and Kato conjecture ζpX , sq to obey a functional equation. Conjecture 2.33 (Functional Equation Conjecture FE(X )). For any complex s one has 2´s s ApX q 2 ζpX , 2 ´ sq “ ˘ApX q 2 ζpX , sq. The lemma below provides special values of the completion factors L8 pH i pXq, sq. We express them in terms of the special values Γ˚ pnq of the Γ-function. Lemma 2.34. One has L8 pH 0 pXq, sq “ ζpS8 , sq “ ΓR psqr ΓC psqs L8 pH 2 pXq, sq “ ζpS8 , s ´ 1q and and, moreover, s“n formula ps ´ 1qr`s ΓC ΓC psqmg´s ζpX8 , sq n ď 0 mpg´1q leading Taylor coefficient at s “ n ´ ¯r`s ` ˘mp1´gq 2π 2p2πq´n Γ˚ pnq n´1 n “ 1 ´pr`sq p2πqr`s π mpg´1q ord s“n ´ L8 pH 1 pXq, sq ΓC psqmg ně2 0 nď0 ´mg ně1 0 2π n´1 ¯r`s ` ˘mp1´gq 2p2πq´n Γ˚ pnq ` ˘mg 2p2πq´n Γ˚ pnq Proof. One checks directly that hij j“0 j“1 hi,˘ ` ´ Γ˚C pnq i“0 m gm i“0 r`s s nď0 i“1 gm m i“1 r`s s ně1 From here the claim is straightforward. 2p2πq´n Γ˚ pnq ord ΓC s“n ´1 0 54 In particular, we may read (2.53) as a decomposition of ζpX8 , sq into factors corresponding to both motivic and cohomological degrees i “ 0, 1, 2. We may consequently define the p completion of each perverse L-function LpH i pX q, sq separately: p p LpH i pX q, sq :“ L8 pH i pXq, sq LpH i pX q, sq. Together with (2.52) Conjecture 2.33 decomposes entirely into motivic degree components and since the functional equation for ζF psq is well-known we arrive at Corollary 2.35. Conjecture 2.33 is equivalent to 1 2´s p ApX q 2 s p LpH 1 pX q, 2 ´ sq “ ˘ 1ApX q 2 LpH 1 pX q, sq. 55 Chapter 3 The special value conjectures in their Weil-étale formulation In this chapter we will define and compute the fundamental line ∆pX , nq as well as the correction factor CpX , nq P Qˆ for arithmetic surfaces X . ∆pX , nq will be a copy of Z that – comes with a distinguished trivialization map λ8 pX , nq : R Ñ ∆pX , nq b R. λ8 pX , nq gives rise to a unique real number up to sign Λ8 pX , nq P Rˆ {t˘1u signifying the inverse generator of the preimage of ∆pX , nq. These quantities feature in the conjectures [8] Conj. 5.10 and 5.11, describing the vanishing orders and leading Taylor coefficients at all integers. We formulate them as follows. Special Value Conjectures. Let X be a proper regular arithmetic surface and let n be any integer. Then one has (VO) ords“n ζpX , sq “ ÿ p´1qi`1 dimR Hci,n pX q iPZ (TC) ζ ˚ pX , nq “ CpX , nqΛ8 pX , nq (3.1) We will explicate these conjectures using the decompositions into motivic degrees of the various cohomology groups worked out in the last chapter. In particular, for n “ 1 the above will turn out to be equivalent to the Birch and Swinnerton-Dyer conjecture. This extends the two-dimensional case of the result [8] Thm. 5.26 — i.e. the compatibility of the above conjectures for smooth projective arithmetic surfaces X with the Tamagawa Number Conjecture — to (not necessarily smooth) proper regular arithmetic surfaces. We keep the notations from Chapter 2. 56 3.1 Conjectural vanishing orders and compatibility with BSD In this section we will explicate [8] Conj. 5.10 for arithmetic surfaces and show compatibility with the rank part of the Birch and Swinnerton-Dyer conjecture. The Vanishing Order Conjecture. We will formulate the vanishing order conjecture as in [8]. Define r RΓar,c pX , Rpnqq “ RΓc pX , Rpnqq ‘ RΓc pX , Rpnqqr´1s i,n r and write Har,c pX q :“ H i pRΓar,c pX , Rpnqqq for its associated cohomology groups. We wish to verify Conjecture 3.1 (Vanishing Order Conjecture VO(X ,n)). For all n P Z one has ords“n ζpX , sq “ ÿ i,n p´1qi ¨ i ¨ dimR Har,c pX q “ iPZ ÿ p´1qi`1 dimR Hci,n pX q. iPZ We begin with a preparational Lemma. We write Λp for the abelian group generated by the irreducible components of Xp modulo the relation rXp s “ 0, i.e. modulo the special fiber Xp interpreted as the weighted sum of its irreducible components. In other words, Λp “ CH0 pXp q{rXp s. Lemma 3.2. We have a short exact sequence à 0 ÝÑ ClF ‘ Λp ÝÑ Pic X ÝÑ Pic X ÝÑ 0. (3.2) p bad 0 X Moreover, since 1H 2,1 pX q – Pic ClF , the above can be rewritten as 0 ÝÑ à Λp ÝÑ 1 H 2,1 pX q ÝÑ Pic0 X ÝÑ 0 (3.3) p bad Proof. The theory of Chow groups provides the Localization Sequence v CH1 pX, 1q ÝÑ à CH0 pXp q ÝÑ CH1 pX q ÝÑ CH1 pXq ÝÑ 0. p ˆ We have CH1 pX, 1q – H 0 pX, OX q “ F ˆ and v is the valuation map. Therefore, after taking À the quotient with the image of v the above sequence becomes (3.2). Since ClF ‘ p bad Λp maps into Pic0 X (cf. e.g. [26] Thm. (8.1.2)(i)) we also have the short exact sequence 0 ÝÑ ClF ‘ à Λp ÝÑ Pic0 X ÝÑ Pic0 X ÝÑ 0. p bad For the second part of the claim take the quotient with ClF . 57 Proposition 3.3. The Vanishing Order Conjecture for X is equivalent to $ ’ mp1 ´ gq for n ă 0 ’ ’ ’ ’ ’ for n “ 0 ’ & mp1 ´ gq ´ 1 ords“n ζpX , sq “ r ` s ´ 1 ´ rk Pic X for n “ 1 ’ ’ ’ ’ ´1 for n “ 2 ’ ’ ’ % 0 for n ą 2 or to $ ’ for n ď 0 ’ & mg 1 0 ords“n LpH pXq, sq “ rk Pic X for n “ 1 ’ ’ % 0 for n ě 2 (3.4) (3.5) In particular, VO(X ,1) is equivalent to the vanishing order part of the Birch and SwinnertonDyer conjecture. Moreover, VO(X ,n) holds for n ě 2. Also, VO(X ,n) is compatible with the conjectural functional equation for LpH 1 pXq, sq. In particular, for an elliptic surface X “ E defined over Z (i.e. S “ Spec Z) one knows VO(E,n) for all n ‰ 1. Proof. The equivalence of (3.4) to VO(X ,n) is immediate from table (A.15). We will now show equivalence to (3.5). Proposition/Definition 2.1 gives ords“n ζpX , sq “ ords“n ζHW pX, sq ´ ords“n ΠpX , sq. For n ‰ 1 Lemma 2.2(i) shows ords“n ΠpX , sq “ 0. Hence – for n ‰ 1 – (3.5) follows from (3.4) when using ζHW pX, sq “ ζF psqζF ps ´ 1q LpH 1 pXq, sq and the well-known formula $ ’ n r ` s ’ ’ ’ & r`s´1 ords“n ζF psq “ ’ ´1 ’ ’ ’ % 0 if n ă 0 if n “ 0 (3.6) if n “ 1 if n ą 1 for the vanishing orders of ζF . By Lemma 2.2(ii) it remains to show ÿ rk Pic X “ rk Pic X ` pdppq ´ 1q . (3.7) p bad This in turn follows from the short exact sequence (3.3) since dppq ´ 1 “ rk Λp . For the second part note that ords“n LpH 1 pXq, sq “ 0 for n ě 2 since the infinite product expression for LpH 1 pXq, sq converges for Repsq ą 3{2. Assuming the motivic degree 1 part of the conjectural functional equation 1 s p 2´s p ApX q 2 L8 pH 1 pXq, sq LpH 1 pX q, sq “ 1ApX q 2 L8 pH 1 pXq, 2´sq LpH 1 pX q, 2´sq (3.8) 58 and using the vanishing order part of Lemma 2.34 verifies ords“n LpH 1 pXq, sq “ mg for n ď 0. The last statement follows since (3.8) is known for X being an elliptic curve over Q by virtue of the Modularity Theorem. 3.2 l Integral Structures In the next two sections we present preparatory results for the computation of the fundamental line ∆pX {Z, nq in Section 3.4 and introduce notation for the integral structures that are left implicit in its definition. An overview of their bases can be found in Appendix A.7. We reserve v|8 for infinite places of F and let j “ 1, . . . , g. From now on we write MB1 ,B2 pf q for the matrix describing a linear map f between real vectorspaces with specified bases B1 and B2 . Integral structures coming from H 1 pX pCq, Cq. The ˘1-eigenspaces H 1 pX pCq, Zq˘ of the GR -action on H 1 pX pCq, Zq induced by the GR -action on X pCq sum to a subgroup of H 1 pX pCq, Zq of finite index H 1 pX pCq, Zq` ‘ H 1 pX pCq, Zq´ Ă H 2 pX pCq, Zq. (3.9) Consequently, their images under the base change maps ´bR 1 H 1 pX pCq, Zq˘ ÝÝÝÝÑ H 1 pX pCq, Rq˘ into the summands of the eigenspace decomposition H 1 pX pCq, Rq “ H 1 pX pCq, Rq` ‘ H 1 pX pCq, Rq´ are integral lattices of maximal order. Since H 1 pX pCq, Zq has a Hodge Decomposition after a base change to C both groups H 1 pX pCq, Zq˘ must have the same rank mg. In fact, Poincaré duality provides a pairing of H 1 pX pCq, Zq with itself that restricts to a pairing between complementary eigenspaces ^ : H 1 pX pCq, Zq` ˆ H 1 pX pCq, Zq´ ÝÑ H 2 pX pCq, Zq – à Z, (3.10) v ` which in turn is a direct sum of perfect pairings over all infinite places of F . Write B ` “ tδvj uvj ´ for a basis of the image of H 1 pX pCq, Zq` in H 1 pX pCq, Rq` and let B ´ “ tδvj uvj “ pB ` q˚ be the basis of H 1 pX pCq, Rq´ dual to B ` . Note that B ` Y B ´ is an R-basis of H 1 pX pCq, Rq but 59 does not necessarily generate the lattice H 1 pX pCq, Zq Ă H 1 pX pCq, Rq since (3.9) is generally not an equality1 . Next, define # ` p2πiqn δvj for n even `;n δvj “ ´ n p2πiq δvj for n odd # and ´;n δvj “ ` p2πiqn δvj for n odd ´ p2πiqn δvj for n even ˘,n and let B ˘;n :“ tδvj uvj Ă H 1 pX pCq, Cq. 1,n pX qR “ H 1 pX pCq, RpnqqGR that generates the integral lattice B `;n is a basis of HW,8 1,n 1,n 2,n HW,8 pX q Ă HW,8 pX qR . Moreover, the set B `,n´1 is for n ě 2 a basis of HD pX q “ 1,n´1 pX q “ H 1 pX pCq, Rpn ´ 1qqGR . H 1 pX pCq, Rpn ´ 1qqGR and for n ď 1 it is a basis of HD Their integral structures shall be the Z-lattices generated by B `;n´1 . Further, for n ď 0 we 1,n endow Hc2,n pX q with an integral structure via Hc2,n pX q – HD pX q. Finally, for any fixed integer n we give H 1 pX pCq, CqGR an integral structure via H 1 pX pCq, CqGR “ H 1 pX pCq, RpnqqGR ‘ H 1 pX pCq, Rpn ´ 1qqGR i.e. BCn :“ B `;n Y B `;n´1 is its integral basis. Integral structures coming from H 2 pX , Zpnqq for n ě 2. For each n ě 2 we fix a set of generators n C n “ tcvj | v|8, 1 ď j ď gu i,n pX qR for these n, the of the image of H 2,n pX q in H 2 pX , ZpnqqR . Since H i,n pX qR – HW i,n set C n also determines an integral structure on HW pX qR . Further, (2.34) shows that 3,2´n 3,2´n 2´n HW pX qR – HompH 2,n pX q, Rq for n ě 2. We write cvj P HW pX qR for the cycle class 2 under this isomorphism. C 2´n :“ tc 2´n | v|8, 1ďjďgu is a basis of the corresponding to cvj vj i,n i,n integral structure of HW pX qR determined by HW pX q since the duality (2.34) is a duality of integral lattices. `;‚ ‚ to refer to integral elements of If there is no risk of confusion we will also use δvj and cvj 2,n 3,2´n Hc2,n pX q for n ď 0 and HW pX q, HW pX q for n ě 2 respectively. Integral structures for de Rham cohomology. One has the decomposition 1 HdR pX pCq{CqGR – H 1 pX , OX qR ‘ H 0 pX , ωX qR . Serre duality for coherent sheaves provides a perfect pairing ^ : H 1 pX , OX q ˆ H 0 pX , ωX q ÝÑ H 1 pX , ωX q – Z 1 (3.11) E.g. if X is an elliptic curve over Q so that one may write XpCq – C{Λ, the matrix B describing the ` 1 ˘ base change from H 1 pX pCq, Zq to B is given by B “ p 1 1 q or B “ 11 ´1 respectively, depending on whether `1 ˘ 1 complex conjugation acts on a basis of the lattice Λ by ´1 or p 1 q. 60 ´b 1 R 10 “ tω u 1 of free abelian groups. Let BdR vj vj be a basis of the image of H pX , OX q ÝÑ 01 :“ pB 10 q˚ :“ tη u 0 10 H 1 pX , OX qR and let BdR vj vj be the basis of H pX , ωX q dual to BdR . We dR will use 10 01 BdR :“ BdR Y BdR “ tωvj , ηvj uvj . A subtlety needs to be taken into account for the choice of bases of derived de Rham cohop2q;n mology groups. Let Brn and Br be bases of the images of H 1,n pX {Zq and H 2,n pX {Zq ddR ddR ddR ddR p2q 1 pX {Rq and H 2 pX {Rq under base change to R. Similarly, let B in HdR 8 8 dR dR be a basis of the ´b 1 R 2 pX {Rq. After splitting off the motivic degree 0 component image of H 0 pS, ωF q ÝÑ HdR 8 (2.47) unravels to ` ˘ ˆ ˙ pnq p2q;n p2q idH 2 pX8 {Rq tddR pX q det MBrddR ApX q n´1 ,BdR ` ˘ “ . ¨ pnq ApSq t pSq det MBrn ,BdR idH 1 pX8 {Rq ddR (3.12) ddR Due to the ad-hoc Definition 2.31(i) the motivic degree 1 and 2 parts of the quotient (2.47) i pX {Rq for i “ 1, 2 can only be expressed in terms of differently defined integral bases of HdR 8 p2q,n n ,B – which we will denote BddR ddR . They have to be chosen in such a way that (2.47) holds p2q;n for each motivic degree component separately. Concretely, we let BddR be any basis of 2 pX {Rq, satisfying HdR 8 ` ˘ detZ RΓddR pS{Zq{F n´1 pn´1q tddR pSq ¨ det MBp2q;n ,Bp2q idH 2 pX8 {Rq “ detZ RΓpS, ωF q ddR dR n 1 pX {Rq that (3.12) remains valid when and then choose BddR to be such a basis of HdR 8 p2q,n p2q,n n n replacing BrddR , BrddR with BddR , BddR . Now the 1-part of (2.47) is ´ ¯ 1 pnq 1 n´1 n tddR pX q ¨ det MBddR id . (3.13) ,BdR H 1 pX8 {Rq “ ApX q dR n n Obviously, if we had a duality result of the kind (2.48) one could choose BddR “ BrddR and p2q;n p2q;n B “ Br . In any case, the difference will not concern us in the remainder of this dR dR thesis. The Period Isomorphism. Let – 1 Φ : H 1 pX pCq, Cq ÝÑ HdR pX pCq{Cq 1 pX {Rq for its restriction be the period isomorphism and write ΦGR : H 1 pX pCq, CqGR Ñ HdR 8 to the GR -invariant part. Lemma 3.4. One has det MB1 ,BdR pΦGR q “ 1. Consequently, for all integers n, det MBn ,BdR pΦGR q “ p2πiq2mgpn´1q . (3.14) 61 Proof. Most of the work towards the above identity is hidden in the definitions of B 1 “ 10 Y B 01 as self-dual bases. It suffices to show that p2πiqB ´ Y B ` and BdR “ BdR dR det MB` YB´ , B01 YB10 pΦGR q “ p2πiq´mg . dR dR In other words, we need to calculate the quantity c P Rˆ {t˘1u for which ľ2gm ľ2gm ľ2gm 1 ΦGR : H 1 pX pCq, CqGR ÝÑ HdR pX8 {Rq (3.15) acts as ľ ´ ` ´ δvj ^ δvj ¯ ÞÝÑ c ¨ ľ v|8 v|8 1ďjďg 1ďjďg pωvj ^ ηvj q . (3.16) ` ´ By the Poincaré and Serre dualities (3.10) and (3.11) δvj ^ δvj and ωvj ^ ηvj are generators ‚ q, i.e. they correspond to classes in of (the v-component of) H 2 pX pCq, Zq and H2 pX , ωX {Z 2 pX {Rq represented by a point. But it is well-known that the period H 2 pX pCq, CqGR and HdR 8 isomorphism on second cohomology Φ : H 2 pX pCq, CqGR ÝÑ H 2 pX8 {Rq is just multiplication with p2πiq´1 (for each v) with respect to point class bases. Consequently, one has c “ p2πiq´mg . Finally, note that the period isomorphism restricts to a map – Φ10 : H 1 pX pCq, Rp1qqGR ÝÑ H 1 pX , OX qR . We define ΩpX q :“ det MB`;1 ,B10 pΦ10 q. dR The duality isomorphism hBpX ,nq . Let hBpX ,nq : H i,n pX qR ÝÑ Hc4´i,2´n pX q˚ be the – piq isomorphism induced by the conjectural perfect pairing (2.29). It is related to the Beilinson 2,n regulator map ρ2 : H 2,n pX qR Ñ HD pX q as follows. Lemma/Definition 3.5. Let n ě 2. Then one has p2q det MC n ,B`;2´n phBpX ,nq q “ det MC n ,B`;n´1 pρ2 q. We write Rn pX q for the above determinants and call it the n-th regulator of X . piq Proof. The hBpX ,nq fit into a commutative diagram (cf. [8] Rmk. 2.6) / / Hci,n pX q p4´iq – phBpX ,2´nq q˚ ρ˚  / H 4´i,2´n pX q˚ R H i,n pX qR ρ / piq / Hc4´i,2´n pX q˚  / H 3´i,2´n pX q˚ D / Hci`1,n pX q p3´iq – hi,n D – hBpX ,nq  / i,n HD pX q – phBpX ,2´nq q˚ ρ˚  / H 3´i,2´n pX q˚ R / 62 i,n 3´i,2´n with exact rows coming from (2.27) and where hi,n pX q is the isomorD : HD pX q – HD phism induced by the perfect pairing (2.23) of Deligne cohomology groups. Specializing to i “ 2 yields the commutative square H 2,n pX qR ρ2 – / H 2,n pX q D p2q – h2,n D – hBpX ,nq  Hc2,2´n pX q˚ (3.17) –  / H 1,2´n pX q˚ D (2.23) simplifies for i “ 2 to the restriction of the Poincaré duality pairing of algebraic topology H 1 pX pCq, Rpn ´ 1qq ˆ H 1 pX pCq, Rp2 ´ nqq ÝÑ H 2 pX pCq, Rp1qq. to its GR -equivariant part. Since Poincaré duality also holds integrally h2,n D does not contribute 2,n pX q Ñ H 1,2´n pX q˚ of to the determinant of the upper right decomposition h2,n R D ˝ ρ2 : H D the square. Finally, since Hc2,2´n pX q derives its integral structure from the bottom map of (3.17) the claim follows from taking determinants in (3.17). 3.3 The Regulator RpX q In this section we will revisit Conjecture 2.18 and introduce the additional assumption that B(X ,n) specializes to the Arakelov intersection pairing. We then define the regulator RpX q of an arithmetic surface X and compare it to the classical regulator RpXq of the generic fiber X. The main result is Proposition 3.11. The pairing B(X ,n). Let σ be a section of the inclusion τ ď2n´1 RΓD pX {R , Rpnqq Ñ r , Rpnqq as the mapping fiber of its composition σ ˝ ρ with RΓD pX {R , Rpnqq. We define RΓpX r i,n pX q “ H i pRΓpX r , Zpnqqq. By the work of last chapter Beilinson’s regulator map. Write H we have a decomposition into motivic degrees p r , Rpnqq » RΓpS, r r , Rpnqqr´1s ‘ RΓpS, r RΓpX Rpnqq ‘ R1 ΓpX Rpn ´ 1qqr´2s. (3.18) r , Rpnqq fits into the 9-Lemma diagram [8] (29) in Section 2.3 whose associated long RΓpX r i,n vanishes for i ‰ 2n and that one has exact sequences show that H r 0,0 pX q – H r 0,0 pSq “ H 0,0 pSqR – R, H r 4,2 pX q – H r 2,1 pSq “ H 2,1 pSqR – R H c (3.19) as well as the short exact sequence r 2,1 pX q ÝÑ H 2,1 pX qR ÝÑ 0. 0 ÝÑ cokerpρ1 q ÝÑ H (3.20) 63 r 2,1 pX q is given by It shows that the motivic decomposition of H ˆ r 2,1 pX q “ cokerpρ1 q ‘ H Pic0 X ClF ˙ ‘ R. (3.21) R r 2n,n pX q is isomorphic to the n-th Arakelov Chow group CHn pX qR (cf. [8] Prop. 2.11) and H the pairing B(X ,n) translates into a perfect pairing r 2n,n pX q ˆ H r 4´2n,2´n pX q ÝÑ R. H (3.22) r 2n,n pX q – CHn pX qR results from an Remark 3.6. Flach’s and Morin’s identification H r 2,1 pX q application of the 5-Lemma and thus depends on the choice of a splitting H 2,1 pX qR Ñ H of (3.20). However, the identifications (3.19) and the decomposition (3.21) into motivic degree r 2n,n pX q – CHn pX qR canonically. components provide one such splitting, i.e. we have H From now on we will assume the following enhanced version of B(X ,n). Conjecture 3.7 (B(X ,n)). Conjecture B(X ,n) holds and the perfect pairing CHn pX qR ˆ CH2´n pX qR ÝÑ R r 2n,n pX q – CHn pX qR coincides with obtained from (3.22) via the canonical identifications H the Arakelov Intersection Pairing x´, ´yAr . For arithmetic surfaces x´, ´yAr is only non-trivial if n ‰ 0, 1, 2. Moreover, it only involves information from motivic degree 1 if n “ 1. We now make the Arakelov pairing explicit for n “ 1 and compare it to the intersection pairing x´, ´yX from algebraic geometry, following [15]. Proposition 3.8. (Arakelov, Hriljac) r 2,1 pX q, i.e. one has (i) (3.21) is an orthogonal decomposition of H ˆ r 2,1 pX q “ cokerpρ1 q K H Pic0 X ClF ˙ K R R and x´, ´yAr is defined on each summand separately. x´, ´yAr is negative definite on pPic0 X {ClF qR (cf. [15] Thm. 3.4, Prop. 3.3). (ii) Let D, D1 P Pic0 X be fibral divisor classes with support in the special fiber Xp . Then xD, D1 yAr “ log N p ¨ xD, D1 yX (cf. [15] def. of pD ¨ Eqv in Sec. 2). 64 (iii) There is a unique linear splitting Pic0 X ÝÑ pPic0 X qQ , P ÞÑ P of the natural projection Pic0 X Ñ Pic0 X such that the image is orthogonal to all fibral divisor classes in Pic0 X (cf. [15] Thm. 1.3). – (iv) Fix an isomorphism φ : Pic0 X ÝÑ Jac X. X has a divisor such that its associated canonical height h satisfies for all P P Pic0 X (cf. [15] Thm. 3.1) xP, PyAr “ ´hpφpP qq. Definitions of RpX q, RpXq and cp pXq. Let P be a basis of the image of 1H 2,1 pX q in H 2,1 pX qR – pPic0 X {ClF qR . P also defines an integral basis on Hc2,1 pX q due to Hc2,1 pX q – 1 H 2,1 pX qR . Let P ˚ be the basis of 1H 2,1 pX q˚R dual to P. We define the regulator RpX q of X to be ´ ¯ ` ˘ p2q RpX q :“ det xP , P 1 yAr P ,P 1 PP “ det MP,P ˚ phBpX ,1q q˚ . Next, fix a basis P “ tPi u1ďiďrk Pic0 X of the image of Pic0 X in pPic0 XqR . The regulator RpXq of the generic fiber X equals ` ˘ RpXq “ det xP, P 1 yAr P,P 1 PP since the Arakelov Intersection Pairing is by Proposition 3.8(iv) the same as the Neron-Tate height pairing on Pic0 X . Now, fix a prime p. Let J Ñ Sp denote the Neron model of the Jacobian J “ Jac XFp of the generic fiber of the local surface XOp over Sp “ Spec Op . Let J˜ “ Jp denote the special fiber of J and let J˜0 be its identity component. We also write J 0 Ă J for the subgroup scheme with generic fiber J and special fiber equal to J˜0 . We define cp pXq :“ # J pFp q #J˜pkppqq . “ J 0 pFp q #J˜0 pkppqq Decomposition of RpX q. Fix a prime p of O. Recall the notations dppq and nj ppq from Lemma 2.2. Also, let tCjp u1ďjďdppq be the reduced irreducible components of Xp and let mj ppq be the multiplicity of Cjp in Xp . The section s : S Ñ X provides a rational point on p p one component – say Cdppq – of Xp . Thus Cdppq must be simple and cannot decompose further over any algebraic extension of kppq, i.e. mdppq ppq “ ndppq ppq “ 1. We conclude that the set of classes Dp :“ trCjp s P pΛp qR u1ďjădppq is a basis of the image of Λp inside pΛp qR . 65 Lemma 3.9. (Raynaud; Bosch, Liu) Fix a prime p of O. The sequence α CH0 pXp q ÝÝÝÝÑ β CH0 pXp q ÝÝÝÝÑ Z dppq ÿ Cjp (3.23) xCip , Cjp yX ¨ Cip ÞÑ i“1 Cjp mj ppq ÞÑ is a chain complex and one has dppq ź Ker β # nj ppq. “ cp pXq ¨ Im α j“1 Proof. This is [5] Theorem 1.11 applied to the abelian variety A “ J . Indeed, the right-most term in Thm 1.11 qdZ{d1 Z vanishes since Xp has a component satisfying mdppq ppq “ ndppq ppq “ 1. Moreover, the geometric multiplicities ej of Cjp in Cjp (cf. [6] Def. 9.1.3) occurring in [5] Rmk. 1.12 equal 1 since the base change of any reduced curve over a perfect field to its algebraic closure remains reduced (see e.g. [26] example (6.1.7)). Remark 3.10. Raynaud has shown the analogue of Lemma 3.9 for algebraically closed residue fields (cf. [26] Prop. 8.12); Bosch and Liu extended it to more general residue fields. As part of his proof Raynaud has shown that, in the case kppq “ kppq, the Picard-scheme Pic0Xp is isomorphic to the group of components J˜{J˜0 of J – a finite étale group scheme that β only depends on the generic fiber X. This should serve as intuition for why # Ker Im α does not depend on the special fiber Xp beyond the values of the nj ppq. Proposition 3.11. One has RpX q “ ˘ “˘ 1 p#Tor Pic0 Xq2 ˆ 1 p#Tor Pic0 Xq2 ˆ Tor Pic X ClF ˙2 Tor Pic0 X ClF ˙2 0 # # ¨ ¨ RpXq ¨ ź ˝plog N pqdppq´1 p dppq ź ˛ nj ppq‚cp pXq j“1 ¨ RpXq ¨ Π˚ pX , 1q ¨ ź cp pXq. p Proof. (3.3) gives a short exact sequence of real vectorspaces ˆ 0 ÝÑ ˙ à p Λp ˆ ÝÑ R Pic0 X ClF ˙ ÝÑ ` ˘ Pic0 X R ÝÑ 0. (3.24) R Due to the splitting provided by Proposition 3.8(iii) the above sequence yields an orthogonal decomposition ˆ Pic0 X ClF ˙ ˆ – R ˙ à p ` ˘ K Pic0 X R Λp R (3.25) 66 and we may regard P 1 :“ P Y Ť pD p as a further (R-)basis of 1H 2,1 pX q . (3.25) gives R ` ˘2 ` ˘ RpX q “ det MP,P 1 pidq ¨ det xP , P 1 yAr P ,P 1 PP 1 ´ ¯ ź ` ˘2 ` ˘ det xrCip s, rCjp sy “ det MP,P 1 pidq ¨ det xP, P 1 yAr P,P 1 PP ¨ 1ďi,jădppq p (3.26) ´ ¯ ` ˘2 ź “ RpXq ¨ det MP,P 1 pidq ¨ plog N pqdppq´1 det xCip , Cjp yX 1ďi,jădppq p where the last equation uses Proposition 3.8(ii). We evaluate the remaining factors separately. First, since (3.3) is an integral exact sequence, det MP,P 1 pidq measures the discrepancy in torsion between Pic0 X {ClF and its surrounding terms in (3.3), i.e. one has det M P,P 1 ´ ¯ id 1H 2,1 pX qR “ 1 Tor Pic0 X ¨# . 0 ClF #Tor Pic X Finally, recall the sequence (3.23) of the previous Lemma. Since mdppq “ 1 the set Dp may also ´ be viewed ¯ as a basis of Ker β. Besides, α is represented by the full intersection matrix p p xCi , Cj yX . It follows that 1ďi,jďdppq ´ ¯ det xCip , Cjp yX 1ďi,jădppq “# Ker β . Im α Lemma 3.9 completes the proof. 3.4 The Fundamental Line Consider the perfect 3ˆ3-square from [8] Prop. 4.14. RΓdR pX8 {Rq{Filn r´1s / RΓD pX {R , Rpnqq / RΓW pX8 , ZpnqqR RΓdR pX8 {Rq{Filn r´1s  / RΓc pX , Rpnqqr1s ‘ RΓc pX , Rpnqq  / RΓW,c pX , ZpnqqR r1s  RΓpX , Rpnqqr1s ‘ RΓpX , Rp2´nqq˚ r´4s – (3.27)  / RΓW pX , ZpnqqR r1s The middle horizontal triangle is exact by the 9-Lemma. So, the fundamental line ∆pX , nq :“ detZ RΓW,c pX , Zpnqq b detZ RΓddR pX {Zq{F n has a trivialization – λ8 pX , nq : R ÝÑ ∆pX , nq b R. By the work of the previous chapter the determinants of the complexes in diagram (3.27) decompose entirely into motivic degree components in the presence of a section s : S Ñ X , 67 satisfying FPB(s,n). So, we may write ∆pX , nq “ ∆pS, nq b 1∆pX , nq´1 b ∆pS, n ´ 1q as well as λ8 pX , nq “ λ8 pS, nq b 1λ8 pX , nq´1 b λ8 pS, n ´ 1q, (3.28) where 1∆pX , nq and 1Λ8 pX , nq denote the Z-line and its trivialization coming from the h 1 -part of the cohomology diagram induced by p3.27q. We write Λ8 pX , nq P Rˆ {t˘1u for the inverse of the generator of the inverse image of ∆pX , nq under λ8 pX , nq, i.e. λ8 pX , nqpZq “ Λ8 pX , nq ¨ ∆pX , nq. Let 1Λ8 pX , nq and Λ8 pS, nq be defined analogously. (3.28) translates into Λ8 pX , nq “ Λ8 pS, nqΛ8 pS, n ´ 1q . 1 Λ8 pX , nq Flach and Morin have worked out Λ8 pS, nq in [8]. Theorem 3.12. (cf. [8] equ. (92) following Prop. 5.33) For n ě 1 write ¯¯ ´ ´ ρn 1,n pSq , Rn pSq “ vol coker H 1,n pSq ÝÑ HD 1,n where the volume is taken with respect to the integral structure of HD pSq coming from 1,n HD pSq – H 0 pFC , p2πiqn´1 ZqR . Then (i) Λ8 pS, nq “ #TS2,1´n #TS1,1´n (ii) Λ8 pS, nq “ 2 p´1qn´1 r ¨ R1´n pSq mn´rn ´s p2πq for n ď 0 ¨ |DF | 1 ´n 2 ¨ #TS2,n #TS1,n ¨ Rn pSq for n ě 1 The remaining part of this section will be dedicated to the proof of the below analogue of Theorem 3.12 for (the motivic degree 1 part of) arithmetic surfaces. Theorem 3.13. With the notations from above and the preceding sections (as well as Appendix A.2 for 1lpX q), one has (i) 1 1 Λ8 pX , nq “ 2n lpX q (ii) # 1TX3,2´n ¨R # 1TX2,2´n ¨ # 1TX4,2´n Tor Pic0 X ΛpX , 1q “ #XpX{F q # ClF ˆ 1 2´n pX q for n ď 0 ˙´2 RpX q ΩpX q ¨ ˛ dppq ź #XpX{F q ΩpX q RpXq ź ˝ “ plog N pqdppq´1 nj ppq‚cp pXq p#Tor Pic0 Xq2 p j“1 68 (iii) Suppose that the technical condition RP(X ) (or the formula (2.39)) holds. Then ˆ ˙n´1 # 1T 3,n p2πq2mg 1 1 Λ8 pX , nq “ 2n lpX q 1 2,n X 1 4,n ¨ ¨ Rn pX q for n ě 2 1 ApX q # T ¨# T X X Remark 3.14. An alternative expression for 1ΛpX , 1q in terms of Br X instead of XpX{F q can be obtained using the comparison sequence induced by (2.26). When defining r´1 ď c ă lpX q via the equation ˆˆ 2 lpX q´1´c “ #Ker Tor Pic0 X ClF ˙_ ÝÑ H then Tor Pic0 X # ClF ˆ 1 ´c ΛpX , 1q “ 2 #Br X Computing ˙ 1 4,1 pX q ˙´2 RpX q ΩpX q. 1 Λ8 pX , nq effectively means to compare the two integral structures on RΓW,c pX , ZpnqqR , one coming from the vertical and one coming from the horizontal distinguished triangle in (3.27) passing through RΓW,c pX , ZpnqqR . (2.36). To do this we will provide explicit descriptions of the maps occuring in the long exact sequences associated to (the h 1 -part of) (3.27) in terms of the integral bases specified in the preceding two sections. Contribution from 2-torsion. Before considering each case n “ 1, n ě 2, n ď 0 ‚,n ‚,n separately we evaluate the contribution of the torsion parts of 1HW,8 pX q and 1HW pX q. Note that for S these torsion groups explain the occurrence of the factor 2p´1q n´1 r in the formula for Λ8 pS, nq with n ě 1 since by Corollary A.16(i) and Corollary A.10 $ 2,1´n n r #T p´1qn n` S ’ 2 2 for n ď 0 & ´ ¯ ´1 χ pRΓW pS, Zpnqqq nr #T 1,1´n p´1qn n` S 2 “ 2 2,n n` ’ χ pRΓW pS8 , Zpnqqq % 2p´1qn p 2 n ´1qr #TS1,n for n ě 1 #T S $ #T 2,1´n S ’ for n ď 0 & #T 1,1´n S “ 2,n ’ % 2p´1qn´1 r #TS1,n for n ě 1. #T S We will now evaluate the h 1 -part of the analogue of the quotient above for X . Corollary A.16(ii) and Corollary A.9 show `p ˘ ` ˘ χ R1 ΓW pX , Zpnqq `p ˘ “ χ pR1 ΓW pX , Zpnqq χ R1 ΓW pX8 , Zpnqq $ ´ ¯´2 ’ ’ Pic0 X ’ for n “ 1 & #XpX{F q # TorCl F “ 1 ’ # T 3,n 1 ’ ’ for n ‰ 1. % 2n lpX q 1 2,n X 1 4,n # T X ¨# T X 69 This explains the leading factors in the formulas of Theorem 3.13. The Trivialization Factor for n “ 1 3.4.1 The below diagram depicts the relevant parts of the long exact sequences induced by the h 1 -part of (3.27). It also displays zero-terms such as p 1TXi,n qR if they carry information on the involved integral structures and hence give rise to the numerical value of 1Λ8 pX , nq. ´ Tor Pic0 X ClF ¯˚ R β21 1,1 pX qR HW,8 1 ´ ` Br X ˘˚ Br O R ‘ Pic0 X ClF  / 1H 2,1 pX qR α2 ` 0 ˘ Pic Xcotor R β2 W,c ¯ / – ´ R  ¯ Pic0 X ClF R  2,1 HW,8 pX qR 1 ` 0 ˘ Pic Xcotor R  / 1H 3,1 pX qR α3 W,c p2q ´ Pic0 X ClF ¯˚  phBpX ,1q q˚ ‘ ` Br X ˘ R Br O R / ´ Pic0 X ClF  ¯˚ R ‘ XpX{F qR  3,1 HW,8 pX qR 1  4,1 HW,c pX qR 1 ´ Tor Pic0 X ClF ¯ / R ´  Tor Pic0 X ClF ¯ . R / H 1 pX pCq, OX pCq qGR / H 1 pX pCq, OX pCq qGR 70 The quotient of the alternating product of torsion cardinalities associated to the two sequences ‚,n passing through HW,c pX q equals Tor Pic0 X #XpX{F q # ClF ˆ ˙´2 Tor Pic0 X #XpX{F q # ClF ˆ p1q tddR pX q “ ˙´2 . 2,1 3,1 The integral lattices of 1HW,c pX qR and 1HW,c pX qR characterized by the vertical maps, are generated by P Y B `;1 and P ˚ respectively. α2 acts as the identity on P. β21 is the period isomorphism. Therefore the trivialization factor becomes ˙´2 Tor Pic0 X Λ8 pX , nq “ #XpX{F q # ¨ det MB`;1 ,B10 pβ21 q ¨ det MP,P ˚ pα3 q dR ClF ˆ ˙ ´2 ´ ¯ Tor Pic0 X p2q “ #XpX{F q # ¨ det MB`;1 ,B10 pΦ10 q ¨ det MP,P ˚ phBpX ,1q q˚ dR ClF ˙ ˆ ´2 Tor Pic0 X “ #XpX{F q # ¨ ΩpX q ¨ RpX q. ClF ˆ 1 In light of Proposition 3.11 this may be reformulated as ¨ ˛ dppq ź ź #XpX{F q ΩpX q RpXq 1 ˝plog N pqdppq´1 nj ppq‚cp pXq. Λ8 pX , nq “ p#Tor Pic0 Xq2 p j“1 71 3.4.2 Trivialization Factors for n ě 2 For n ě 2 the h 1 -part of the diagram on cohomology induced by (3.27) becomes p 1TX2,2´n q˚R 1,n HW,8 pX qR 1  2,n HW,c pX qR 1 p 1TX3,2´n q˚R ‘ pZmg ‘ 1TX2,n qR 1,n HdR  – X 0  / 1H 2,n pX qR 0 D W,8 1,n HdR  3,n HW,c pX qR 1 p 1TX4,2´n q˚R ‘ p 1TX3,n qR  / p 1T 3,n qR – X  3,n HW,8 pX qR 1  4,n HW,c pX qR 1  / p 1T 4,n qR p 1TX4,n qR X  4,n HW,8 pX qR . 1 / H 1,n dR β2  / pZmg ‘ 1T 2,n qR ρ2 / H 2,n pX q β21 / H 1,n dR γ21 / H 2,n D 72 2,n The integral structure of 1HW,c pX qR induced by the vertical sequence is generated by Bcn “ B `;n Y C n . The diagram shows that 1 # 1TX3,n pnq n pβ2 q ¨ 1tddR pX q´1 ¨ det MBcn ,BddR 1 2,n 1 4,n # TX ¨ # TX # 1T 3,n 1 1 “ 2n lpX q 1 2,n X 1 4,n ¨ 1 ¨ det MBcn ,BdR pβ2 q ApX qn´1 # TX ¨ # TX ˆ ˙n´1 # 1TX3,n p2πq2mg n 1lpX q “2 ¨ ¨ det MBcn ,Bn pβ2 q 1 ApX q # 1TX2,n ¨ # 1TX4,n ˆ ˙n´1 # 1TX3,n p2πq2mg n 1lpX q ¨ “2 ¨ det MC n ,B`;n´1 pρ2 q 1 ApX q # 1TX2,n ¨ # 1TX4,n ˙n´1 ˆ # 1TX3,n p2πq2mg n 1lpX q ¨ Rn pX q. “2 ¨ 1 ApX q # 1TX2,n ¨ # 1TX4,n n 1lpX q Λ8 pX , nq “ 2 by (3.13) by (3.14) by Def. 3.5 73 3.4.3 Trivialization Factors for n ď 0 Finally, for n ď 0 the h 1 -part of (3.27) equals p 1TX4,2´n q˚R  α12 1,n HD pX q / 1H 1,n pX qR W,8 –    / 1H 2,n pX qR α2 Hc2,n pX q W,c 0 p 1TX4,2´n qR ‘ p 1TX3,2´n q˚R  / p 1T 4,2´n qR – X  2,n HW,8 pX qR 1  / 1H 3,n pX qR α3 Hc2,n pX q W,c p2q  phBpX ,2´nq q˚ p 1TX3,2´n qR ‘ pZmg ‘ 1TX2,2´n q˚R –  / pZmg ‘ 1T 3,2´n qR X  3,n pX qR HW,8 1  4,n HW,c pX qR 1 p 1TX2,2´n qR –  / p 1T 2,2´n qR . X 2,n 3,n The vertical sequence endows 1HW,c pX qR and 1HW,c pX qR with integral structures generated by B `;n and C 2´n . So, since det MB`;n ,B`;n pα2 q “ 1, the diagram shows 1 Λ8 pX , nq “ 2 n 1lpX q # 1TX3,2´n # 1TX2,2´n ¨ # 1TX4,2´n ¨ det MB`;n ,C 2´n pα3 q. The proof of Theorem 3.13 is complete after observing that the commutative square involving α3 shows p2q det MB`;n ,C 2´n pα3 q “ det MB`;n ,C 2´n pphBpX ,2´nq q˚ q “ R2´n pX q. 74 3.5 The Correction Factor Definition of CpX , nq. Geisser has shown that the étale topology of a curve equals the eh-topology of the corresponding reduced curve. Therefore the definition of CpX , nq in [8] Sec. 5.3, 5.4 simplifies for π : X Ñ S as follows. Definition 3.15. For each prime p and n P Z let χpXFp , O, nq “ ÿ p´1qk`j pn ´ kq dimFp H j pXFred , ΩkX red {Fp q. p Fp 0ďkăn, jPZ In particular, χpXFp , O, nq “ 0 for all p whenever n ď 0. Conjecture/Definition 3.16. For any prime p and n P Z one has a distinguished triangle in the derived category of Qp -vectorspaces RΓdR pXQp {Qp q{F n r´1s ÝÑ RΓpXZp , Qp pnqq ÝÑ RΓpXFred , Qp pnqq ÝÑ . p (3.29) In particular, there is a trivialization ¯ ´ ` ˘ – n λp pX , nq : detZp RΓpXZp , Zp pnqq Q ÝÑ detZp RΓpXFred , Zp pnqq b det´1 Zp RΓddR pXZp {Zp q{F p p (3.30) Qp χpXFp ,O,nq ˆ that specifies a power Λp pX , nq “ detZp λp pX , nq P Qˆ ¨ p {Zp of p. We let cp pX , nq “ p Λp pX , nq and define the correction factor for X and n to be CpX , nq “ ź |cp pX , nq|p . pă8 The proof of [8] Prop. 5.9 shows that cp pX , nq is trivial unless p ď n ` 1 or XFp is a bad fiber. Thus, CpX , nq is well-defined. Moreover, one has CpX , nq “ 1 for n ď 0 (cf. [8] Prop. 5.7). The leading Taylor coefficient conjecture. Write ζ ˚ pX , nq for the leading coefficient of the Taylor expansion of ζpX , sq at s “ n. Define ζ ˚ pX8 , nq and ζ ˚ pX , nq analogously. We can now formulate Flach’s and Morin’s conjectural description of ζ ˚ pX , nq (cf. [8] Conj. 5.11). Conjecture 3.17 (Leading Taylor Coefficient Conjecture TC(X ,n)). For any integer n one has ζ ˚ pX , nq “ CpX , nqΛ8 pX , nq. 75 Decomposition of CpX , nq into motivic degrees. Let ÿ i χpXFp , O, nq :“ p´1qi , ΩkX red {Fp q pn ´ kq dimFp H i´k pXFred p Fp 0ďkďn i.e. iχpXFp , O, nq is the sub-summation of those summands of χpXFp , O, nq for which j `k “ i. One has iχpXFp , O, nq “ 0 whenever i ‰ 0, 1, 2 as desired. To define a decomposition ś i Λp pX , nq “ i“0,1,2 iΛp pX , nqp´1q we will proceed as in Definition 2.31 and let X red p 0 X red S red R π˚ Zp pnq Fp :“ R0 π˚ Zp pnq Fp “ Zp pnq Fp and p 2 X red S red R π˚ Zp pnq Fp :“ Zp pn ´ 1q Fp as well as p red detZp R1 ΓpXFred , Zp pnqq :“ detZp τ ě1 RΓpXFred , Zp pnqq b det´1 Zp RΓpSFp , Zp pn ´ 1qq. (3.31) p p Again, the complex RΓpXFred , Zp pnqq does not necessarily decompose since XFred is not p p p necessarily smooth, i.e. the symbol R1 ΓpXFred , Zp pnqq itself is undefined. We introduce p (3.31) to force a splitting on the level of determinants ¯p´1qi â ´ p detZp Ri ΓpXFred , Z pnqq . p p detZp RΓpXFred , Zp pnqq “ p i“0,1,2 Theorem 2.11 provides a motivic decomposition of RΓpXZp , Zp pnqq after passing to the p-adic completion. A decomposition of detZp RΓddR pXZp {Zp q{F n is given in (2.51). So, every term in (3.30) decomposes and we may define iΛp pX , nq as the trivialization factor of the i h i -component of (3.30). Finally, after defining icp pX , nq “ p χpXFp ,O,nq ¨ iΛp pX , nq and i CpX , nq “ ź | icp pX , nq|p pă8 we get the decomposition CpX , nq “ źi CpS, nq CpS, n ´ 1q i CpX , nqp´1q “ . 1 CpX , nq iPZ (3.32) Results for S. Flach and Morin have computed the correction factor for S assuming the validity of a conjecture from p-adic Hodge theory. Proposition 3.18. (cf. [8] Prop. 5.33) When assuming Conjecture CEP pQp pnqq in [24][App.C2] for all local fields Fv of F one has # CpS, nq “ 1 for n ď 1 pn ´ 1q!´m for n ě 1. 76 Computation of CpX , 1q. Fix a rational prime p. For any complex F of sheaves on a scheme X we will use the notation ˆ ˙ ‚ x RΓpX , F q :“ RΓ X , lim F {p “ lim RΓpX , F {p‚ q ÐÝÝ ÐÝÝ to denote p-adic completion2 . Theorem 3.19. For n “ 1 the triangle (3.29) exists. Moreover, CpX , 1q “ 1. Proof. Write X “ XZp and let Z “ XFred . Let ι : Z Ñ X denote the closed immersion of p the reduced special fiber. Write I Ă OX for the ideal sheaf of Z, i.e. there is a short exact sequence 0 ÝÑ I ÝÑ OX ÝÑ OZ ÝÑ 0. (3.33) red over each Note that I is the radical of ppq “ pOX . Z is the disjoint union of fibers Zp “ Xkppq prime p dividing p. Write ppa “ pa pZp q for their arithmetic genera. For n “ 1 the triangle (3.29) becomes Dp p1q : exp x x RΓpXQp , OXQp q ÝÑ RΓpX , Gm q b Qp ÝÑ RΓpZ, Gm q b Qp ÝÑ (3.34) after shifting by one degree since RΓpXQp , OXQp q is p-adically complete already. We will show that the above triangle is exact, and moreover, compute the associated trivialization factor Λp pX , 1q by comparing it to the below distinguished triangle (3.36) of integral lattices. Fix a power p‚ of p. One checks on stalks that 1 ÝÑ p1 ` Iq{p‚ ÝÑ pGm {p‚ qX ÝÑ ι˚ pGm {p‚ qZ ÝÑ 1 (3.35) is a short exact sequence of abelian sheaves on X . Applying the derived global sections functor and then passing to p-adic completions yields x , Gm q ÝÑ RΓpZ, x x , 1 ` Iq ÝÑ RΓpX RΓpX Gm q ÝÑ . (3.36) The two right most complexes in (3.36) coincide with the integral structures of the two right most complexes in (3.34). So, it remains to compare the integral lattice RΓpX , OX q inside RΓpXQp , OXQp q with RΓpX , 1 ` Iq. A diagrammatic overview of this comparison as worked out in the remainder of this proof is given in Remark 3.20 below. Write pX , OX q for the formal completion of X along Z and let ιX : Z Ñ X denote the inclusion into X . Let IX :“ IOX . One has IX “ lim I{I ‚ “ lim I{p‚ . Moreover, the ÐÝÝ ÐÝÝ Theorem on Formal Functions gives for any k ě 0 k RΓpX , I k q » RΓpX , IX q 2 When applied to objects in a derived category lim is understood to mean the homotopy limit. ÐÝÝ 77 since I k is coherent. The transition to formal scheme theory equips us with a logarithm map r – I r for sufficiently log : 1 ` IX Ñ OX which reduces to an isomorphism log : 1 ` IX X large r. Consequently, for some fixed r, one has r r x RΓpX , IX q » RΓpX , 1 ` IX q. Next, consider the short exact sequences k`1 k`1 k 1 ÝÑ 1 ` IX ÝÑ 1 ` IX ÝÑ p1 ` IX qk {p1 ` IX q ÝÑ 1 k`1 k`1 k k 0 ÝÑ IX ÝÑ IX ÝÑ IX {IX ÝÑ 0 for k “ 1, 2, . . . , r ´ 1. One checks directly that k`1 k`1 k k IX {IX ÝÑ p1 ` IX q{p1 ` IX q, f ÞÑ 1 ` f is an isomorphism of sheaves. Therefore we obtain x detZp RΓpX , 1 ` IX q detZp RΓpX , IX q “ , r x detZp RΓpX , IX q detZp RΓpX , 1 ` I r q X where the above should be read as an equality of quotients of Zp -lattices inside the (1x dimensional) Qp -vectorspaces detQ RΓpXQ , IX q and detQ RΓpX Q , 1`IX q respectively. p p p Qp p Qp Moreover, the long exact sequence associated to (3.33) equals 0 0 Ñ ΓpX , Iq Ñ OZp Ñ OFred ÝÑ H 1 pX , Iq Ñ H 1 pX , OX q Ñ p à p 0 kppqpa ÝÑ p|p 0 – ÝÑ H 2 pX , Iq ÝÑ H 2 pX , OX q Ñ 0. So, we obtain p ź p detZp RΓpX , OX q detZp RΓpX , OX q ź #kppqpa “ “ “ N pppa ´1q . detZp RΓpX , IX q detZp RΓpX , Iq #kppq p|p (3.37) p|p It remains to relate the cohomology complexes of p1 ` Iq{p‚ and p1 ` IX q{p‚ . By virtue of the proper base change theorem the canonical morphisms of sheaves on Z φ‚ : ι˚ p1 ` Iq{p‚ ÝÑ ι˚X p1 ` IX q{p‚ induce a compatible system of maps RΓpφ‚ q RΓpX , p1`Iq{p‚ q “ RΓpZ, ι˚ p1`Iq{p‚ q ÝÝÝÑ RΓpZ, ι˚X p1`IX q{p‚ q “ RΓpX , p1`IX q{p‚ q. We will show later that the induced map between p-adic completions is an isomorphism: » x , 1 ` Iq ÝÑ x lim RΓpφ‚ q : RΓpX RΓpX , 1 ` IX q. ÐÝÝ (3.38) 78 It will show that Λp pX , 1q must equal (3.37). This will finish the proof since it combines with ÿ ÿ χpXFp , O, 1q “ p´1qj dimFp H j pZp , OZp q j p|p to the desired result CpX , 1q “ pχpXFp ,O,1q ¨ Λp pX , 1q “ ź ¨ ˛p´1qj ź ź p ˝ #H j pZp , OZp q‚ ¨ N pppa ´1q “ 1. j p|p p|p The proof of (3.38) amounts to the algebraic exercise of verifying that taking mapping cones of the multiplicative sheaf 1 ` I interacts well with transitioning to the formal (i.e. p-adic) completion IX “ lim I{p‚ of the additive sheaf I. Clearly, each φ‚ and hence ÐÝÝ lim φn : lim p1 ` Iq{pn ÐÝÝ ÐÝÝ n lim p1 ` IX q{pn “ lim lim p1 ` I{pm q{pn ÐÝÝ ÐÝÝÐÝÝ ÝÝÝÝÑ n n n m is injective. For surjectivity it suffices to show that for each pair of integers n, m there is an N ě n such that p1 ` Iq{pN ÝÑ p1 ` I{pm q{pn surjects. To do this it is enough to exhibit a constant c such that for all N ě c one has N p1 ` Iqp Ă 1 ` pN ´c I. ř N `p N ˘ k N Recall that for some s one has I s Ă ppq. Let f P I and consider p1 ` f qp “ 1´` pk“1 Y i ]¯k f . `pN ˘ k i k t u Since ordp k ą N ´ i for k ă p and f P ppq s one may choose c “ maxi i ´ ps . Remark 3.20. The diagram below summarizes the comparison between RΓpX , OX q and x , 1 ` Iq made in the proof. Also note that the p-adic completeness of the complexes in RΓpX Dp p1q is essential as otherwise there would be no way to relate RΓpX , 1`IX q to RΓpX , 1`Iq. RΓpX , OX q x , 1 ` Iq RΓpX “ lim RΓpφ‚ q » ÐÝÝ pppa ´1q p|p N p  RΓpX , IX q Ă x RΓpX , 1 ` IX q y detZp RΓpX ,IX q y detZ RΓpX ,I r q p r q RΓpX , IX X y detZp RΓpX ,1`IX q y detZ RΓpX ,1`I r q p log » Ă Ă RΓpX , OX q ś X x / RΓpX , 1 ` I r q. X 79 Remark 3.21. The presented proof does not use dim X “ 2 at any point. So – when recalling the more general definition of the correction factor in [8] – the above proof shows CpX , 1q “ 1 for proper regular arithmetic schemes X of any dimension. Remark 3.22. Since we already know CpS, 1q “ CpS, 0q “ 1 the above shows iCpX , nq “ 1 1 ś p for each i “ 0, 1, 2. One has 1Λp pX , 1q “ p|p pN pqpa which cancels with p χpXFp ,O,1q “ ś ´ppa . p|p pN pq Remark 3.23. We keep the notation of the proof and provide a more geometric version of it for X with good reduction. In this case Dp p1q decomposes entirely into motivic degree components and it suffices to understand its h 1 -part. Write J “ JacX {Zp for the Jacobian variety of X . J is a projective abelian variety of relative dimension g over OZp . Let J ãÑ PN Zp be defined in terms of homogeneous equations in variables T0 , . . . , TN . Let O “ Spec OZp ãÑ J be its unit section. Choose local coordinates X1 , . . . , Xg of J at O, i.e. X1 , . . . , Xg is a set of generators of the maximal ideal m of the ring of regular functions OO :“ pOJ qO at O. Since Z is smooth its Jacobian variety J is obtained from reducing J modulo p, i.e. J “ JF . Write Jˆ and Jˆ for the formal groups of J , J with respect to their p local coordinates tXj u1ďjďg and tX j u1ďjďg . The motivic degree-1-part of the H 1 -groups of the long exact sequence belonging to Dp p1q is given by exp { { red 0 ÝÑ H 1 pX , OX q b Qp ÝÑ J pOq b Qp ÝÑ JpO Fp q b Qp ÝÑ 0. (3.39) Since H 1 pX , OX q – H 0 pX , ωX q˚ where ωX denotes the canonical bundle of X the choice of tXj u1ďjďg corresponds to a choice of an integral basis of H 1 pX , OX q b Qp “ H 1 pXQp , OXQp q p ga pOZ q. and hence lets us identify its integral structure with G p Let p be the Jacobson radical of OFred , i.e. OFred {p “ p p À p|p kppq. We may assume O “ r1 : 0 : ¨ ¨ ¨ : 0s, i.e. Ti P m for all 0 ă i ď N . Then the Ti may be interpreted as power series ‚ y Ti “ T̂i pX1 , . . . , Xg q P O O “ lim OO {m in the variables X1 , . . . , Xg . This gives for any ÐÝÝ exponent e ě 1 a morphism of abelian groups { ι : Jˆpp e q ÝÑ J pp e q, pxj q1ďjďg ÞÑ T̂i px1 , . . . , xg q. p ga that restricts for sufficiently Formal Group Theory provides a logarithm map log : Jˆ ÝÑ G large r ą 0 to an isomorphism of abelian groups – p g pp r q “ log : Jˆpp r q ÝÑ G a à pr . 1ďjďg The exponential map in (3.39) is the base change of the inverse of the above logarithm to Qp . 80 Therefore, the (non-exact) restriction of p3.39q to integral structures fits into the diagram exp p ga pOq /G Ă 0 / J{ pOZp q O { red q / JpO Fp /0 ι log Jˆppq – / Jˆpp r q Ă Ă p ga ppq o G p ga pp r q o G M M p ga pp k q G p ga pp k`1 q – Jˆpp k q Jˆpp k`1 q for all k ě 1, the trivialization Since, moreover, G factor associated to (3.39) does not depend on r and agrees with the trivialization factor of ι { red 0 ÝÑ Jˆppq ÝÑ J{ pOZp q ÝÑ JpO Fp q ÝÑ 0 (3.40) p ga pOq{G p ga ppq “ pś N pqg – which equals ś pN pqppa since we assume up to a factor of #G p|p p|p Z to be smooth. So, it suffices to show that (3.40) is exact. First, let t “ rt0 : t1 : ¨ ¨ ¨ : tN s P J{ pOZp q be an integral point that is congruent to O modulo p. The Xj “ Xj pT0 : ¨ ¨ ¨ : Tn q are rational functions in the Ti that vanish at O. Therefore we have xj :“ Xj ptq P p. We then have ιpxj qj “ t proving exactness at the middle component. Let now Zn “ X ˆ Spec Z{pn . By the Theorem on Formal Functions the categories of line bundles on X and X are equivalent. So, one has J pOZp q “ Pic0 X “ lim Pic0 Zn . Therefore ÐÝÝ it suffices to show that each Pic0 Zn surjects onto J pOFp q “ Pic0 Z. This in turn follows from the long exact sequence associated to Z n 1 ÝÑ 1 ` pOZn ÝÑ GZ m ÝÑ Gm ÝÑ 1 since H 2 pZ, 1 ` pOZn q “ 0 because Z is one-dimensional. Note that exactness of (3.40) in the special case of an elliptic surface is precisely [32] Ch. VII, Prop. 2.1 & 2.2. 3.6 The Functional Equation The correction factor for n ě 2. For higher twists n ě 2 the correction factor CpX , nq may be computed using the conjectural functional equation FE(X ). Theorem 3.24. Assume conjectures FE(X ) and TC(X ,n) hold. Then for n ě 2 one has ˙ ˆ ˚ Γ p2 ´ nq mg 1 “ ppn ´ 1q! ¨ pn ´ 2q!q´mg . (3.41) CpX , nq “ ˘ Γ˚ pnq Similarly, when assuming TC(S,n) one has for n ě 1 CpS, nq “ pn ´ 1q!´m 81 and consequently CpX , nq “ ppn ´ 1q! ¨ pn ´ 2q!qmpg´1q . Proof. Let n ě 2. Corollary 2.35 unfolds to 1 2´n p n p L˚8 pH 1 pXq, 2´nq 1Λ8 pX , 2´nq “ ˘ 1ApX q 2 L˚8 pH 1 pXq, nq 1Λ8 pX , nq 1CpX , nq. ApX q 2 By virtue of Theorem 3.13 and Lemma 2.34 this simplifies to 1 CpX , nq “ ˘ 1ApX q1´n ¨ p ˚ L8 pH 1 pXq, 2 ´ nq 1Λ8 pX , 2 ´ nq ¨ 1 p ˚ L8 pH 1 pXq, nq Λ8 pX , nq ¸´1 ˆ ˙mg ˜ ˚ p2πiq2mgpn´1q 2pn´1q Γ p2 ´ nq “ ˘ ApX q ¨ p2πq 1 Γ˚ pnq ApX qn´1 ˙mg ˆ ˚ Γ p2 ´ nq . “˘ Γ˚ pnq 1 1´n Similarly, for n ě 1 we obtain from the well-known functional equation FE(S) 1 CpS, nq “ ˘ApSq 2 ´n ¨ Γ˚R p1 ´ nqr Γ˚C p1 ´ nqs Λ8 pS, 1 ´ nq ¨ Γ˚R pnqr Γ˚C pnqs Λ8 pS, nq ˜ “ ˘p#DF q 1 ´n 2 π n´ 21 ` 1 ˘´n ˜ 2 Γ˚ p 1´n 2 q Γ˚ p n2 q Γ˚ p 1´n 2 q n ˚ Γ p2q ¸r ˆ p2πq2n´1 Γ˚ p1´nq Γ˚ pnq ˙s 1 2p´1qn´1 r p2πqmn´rn ´s p#D 1 F q2 ´n ¸r pn ´ 1q!´2s m 2p´1qn´1 r p2πq 2 ´rn ´s ¸ ˜ ˚ p 1´n q r p´1qn Γ 1 1 nr ´n` ` r ´ r p´1q 2 nq 2 “ ˘2p p2πq 2 2 Γ˚ p n2 q pn ´ 1q!2s n´ 21 “ ˘2´p qr 2n r ˆ p1´nqr “ ˘2 p´1qn 2n´1 π 2 pn ´ 1q! ˙r 1 pn ´ 1q!2s “ pn ´ 1q!´m . Note that the factor p1{2q´n arises from the relation between leading Taylor coefficients Γ˚R pkq “ p1{2qords“k ΓR psq π ´k{2 Γ˚ pk{2q . (3.32) concludes the proof. In particular, CpS, nq “ pn ´ 1q!´m also holds for n ě 1 when assuming TC(S,n) instead of the conjecture CEP pQp pnqq from p-adic Hodge Theory. A simplified version of FE(X ) for integers n “ s. Flach and Morin provide a reformulation of FE(X ) for integer arguments in terms of a newly defined quantity x8 pX , nq P Rą0 , satisfying x8 pX , nq2 “ Λ8 pX , 2 ´ nq . Λ8 pX , nq (3.42) 82 The definition of x8 pX , nq2 will not incorporate the full fundamental line, mirroring the fact that the quotient on the right hand side is easier to evaluate than each term separately. We will review it here and use it to compute x8 pX , nq independently of Theorem 3.13. Set n Ξ8 pX {Z, nq :“ detZ RΓW pX8 , Zpnqq b det´1 Z RΓddR pX {Zq{F b 2´n det´1 . Z RΓW pX8 , Zp2 ´ nqq b detZ RΓddR pX {Zq{F (3.43) This definition is made to have an isomorphism – φ : ∆pX {Z, nq b Ξ8 pX {Z, nq ÝÑ ∆pX {Z, 2 ´ nq. Observe that the distinguished triangle RΓdR pX8 {Rq{F n r´1s ÝÑ RΓD pX {R , Rpnqq ÝÑ RΓW pX8 , ZpnqqR ÝÑ (3.44) together with the duality (2.23) for Deligne cohomology gives a trivialization Ξ8 pX {Z, nqR » detR RΓD pX {R , Rpnqq b det´1 R RΓD pX {R , Rp2 ´ nqq » detR RHompRΓD pX {R , Rp2´nqq, Rr´3sq b det´1 R RΓD pX {R , Rp2´nqq » detR RΓD pX {R , Rp2´nqq b det´1 R RΓD pX {R , Rp2´nqq (3.45) » R. – We denote it by ξ8 pX , nq : R ÝÑ Ξ8 pX {Z, nqR and define x28 pX , nq P Rą0 via ξ8 pX , nqpZq “ x28 pX , nq ¨ Ξ8 pX {Z, nq as an equality of lattices in Ξ8 pX {Z, nqR . Clearly x8 pX , 2´nq “ x8 pX , nq´1 . One has Proposition 3.25. (cf. [8] Prop. 5.28, Cor. 5.30) The diagram φbR ∆pX {Z, nq b Ξ8 pX {Z, nq b R O / ∆pX {Z, 2 ´ nq b R O λ8 pX ,nq b ξ8 pX ,nq RbR λ8 pX ,2´nq R commutes. Therefore – when assuming TC(X ,n) for all integers n – the functional equation FE(X ) holds for all integers s “ n if and only if for all n 2´n n ApX q 2 ¨ ζ ˚ pX8 , nq ¨ CpX , nq ApX q 2 ¨ ζ ˚ pX8 , 2 ´ nq ¨ CpX , 2 ´ nq “˘ . x8 pX , nq x8 pX , 2 ´ nq (3.46) Motivic decomposition of Ξ8 pX {Z, nq. By the work of last chapter all determinants in (3.43) admit motivic decompositions. So, we obtain further decompositions 1 Ξ8 pX {Z, nq “ Ξ8 pS{Z, nq b Ξ8 pX {Z, nq´1 b Ξ8 pS{Z, n ´ 1q and x8 pX , nq “ x8 pS, nq ¨ 1x8 pX , nq´1 ¨ x8 pS, n ´ 1q. 83 Proposition 3.26. Let n be any integer. One has 1 n x28 pS, nq “ 2p´1q r p2πqrn `s´mn p#DF qn´ 2 . Moreover, when assuming the technical condition RP(X ) (or the formula (2.39)), one has 1 1x2 pX , nq “ 8 and consequently ApX qn´1 p2πq2mgpn´1q ´ ¯n´1 x28 pX , nq “ p2πq2mpg´1q ApX q . Proof. It suffices to consider n ě 1 since x8 pS, 1 ´ nq “ x8 pS, nq´1 and x8 pX , 2 ´ nq “ x8 pX , nq´1 . The second quasi-isomorphism of (3.45) is due to Poincaré Duality which holds integrally; the third and the fourth follow directly from the determinant formalism. Therefore the trivialization factors x28 pX , nq and x28 pS, nq arise fully from a comparison of the integral structures of the complexes in (3.44) (and its analogue for S). The motivic degree 1 part of its associated long exact sequence is 1,n 1,n 2,n 0 ÝÑ HW,8 pX qR ÝÑ HdR pX qR ÝÑ HD pX q ÝÑ 0, which is exact integrally with respect to B n . Consequently, for n ě 2, 1x2 pX , nq “ 1tpnq pX q ¨ det M n BddR ,Bn pidq 8 ddR “ 1 p2πq2mgpn´1q pnq n ¨ 1tddR pX q ¨ det MBddR ,BdR pidq by Lemma 3.4 1 “ ApX qn´1 p2πq2mgpn´1q by (3.13). The 2-torsion of RΓW pX8 , Zpnqq does not contribute due to Corollary A.9. For S we begin noting that by Corollary A.16(i) one has for all n ¯p´1qi ź´ n n`n i,n χpRΓW pS8 , Zpnqqq “ pSq “ 2p´1q 2 r . #Tor HW,8 iPZ The long exact sequence associated to the analogue of (3.44) for S becomes 0,n 0,n 1,n 0 ÝÑ HW,8 pSqR ÝÑ HdR pSqR ÝÑ HD pSq ÝÑ 0. The implied integral structures are obvious. The outer cohomology groups contribute factors of p2πiq´npn r`sq and p2πiq´pn´1qpn r`sq respectively to x28 pS, nq. The relative determinant 0,n between the lattices HddR pS{Zq and H 0 pS, OF q “ OF equals ApSqn´1 “ p#DF qn´1 . A ? remaining factor of #DF results from the comparison of OF with the integral lattice of the Minkowski space FR . Therefore one obtains 2 x8 pS, nq “ 1´n 1´n`1´n r 2 2p´1q nr p´1qn n` 2 2 n p2πiq´npn r`sq´pn´1qpn r`sq p#DF qn´1 1 “ 2p´1q r p2πqrn `s´mn p#DF qn´ 2 a #DF 84 as claimed. Finally, we combine this to x8 pS, nqx8 pS, n ´ 1q 1x pX , nq 8 ¸ ˜ ˆ ˙ ´ ¯n´1 |DF | 2pn´1q p2πq2mgpn´1q 2mpg´1q “ “ p2πq ApX q . ¨ 1 p2πqm ApX qn´1 x28 pX , nq “ In particular, (3.42) holds for every motivic degree component separately. So, we can rederive the formula for the correction factor (3.41) from (3.46). 3.7 Summary of special value results We may now combine Theorem 3.13 and Theorem 3.12 with the results on the correction factor Theorem 3.19 and Theorem 3.24 to obtain closed formulas for the leading Taylor coefficients of ζpX , sq. These are presented in the Theorem below. Moreover, when combining the second equality in Theorem 3.13(ii) with Lemma 2.2(ii) one obtains 1 L˚ pH 1 pXq, 1q “ Λ8 pX , 1q 1CpX , 1q #XpX{F q ΩpX q RpXq ź “ cp pXq, Π˚ pX , 1q p#Tor Pic0 Xq2 p which is precisely the leading Taylor coefficient part of the Birch and Swinnerton-Dyer conjecture for the abelian variety Pic0 X (cf. [35] equ. (1.5)). We may thus summarize the special value results of this chapter as follows. Theorem 3.27. Let the notation be as per this and the preceding chapter. We make the standard assumptions that L(X ,n) and B(X ,n) hold for all integers n. Suppose π : X Ñ S has a section s : S Ñ X satisfying FPB(s,n) for all n ě 2. Then (i) VO(X ,n) is equivalent to $ ’ mp1 ´ gq ’ ’ ’ ’ ’ ’ & mp1 ´ gq ´ 1 ords“n ζpX , sq “ or to for n “ 0 r ` s ´ 1 ´ rk Pic X ’ ’ ’ ’ ´1 ’ ’ ’ % 0 $ ’ ’ & ´1 ´ rk Pic X ords“n ζpX , sq “ for n ă 0 ´1 ’ ’ % 0 for for n “ 1 , (3.47) for n “ 2 for n ą 2 n“1 for |n ´ 1| “ 1 . for |n ´ 1| ą 1 (3.48) 85 (ii) TC(X ,1) is equivalent to ζ ˚ pX , 1q “ p#Tor Pic0 X q2 2r p2πqs RpSq2 ? ¨ ¨ . p#µF q2 #DF #XpX{F q ¨ ΩpX q RpX q For n ď 0, TC(X ,n) is equivalent to ˚ pr´lpX qqn ζ pX , nq “ 2 #TX2,2´n ¨ #TX4,2´n R2´n pSqR1´n pSq ¨ . R2´n pX q #T 1,2´n ¨ #T 3,2´n X X We now suppose that the technical condition RP(X ) (or the formula (2.39)) holds. When further assuming FE(X ), the pair of conjectures TC(X ,n) and TC(X ,2 ´ n) implies # CpX , nq “ 1 for n ď 1 ppn ´ 1q! ¨ pn ´ 2q!qmpg´1q for n ě 2 . (3.49) When assuming FE(X ) and (3.49) then, for n ě 2, TC(X ,n) is equivalent to ˚ pr´lpX qqn ζ pX , nq “ 2 #TX2,n #TX4,n #TX1,n #TX3,n ˆ pn ´ 1q!pn ´ 2q! p2πq2pn´1q ˙mpg´1q ApX q1´n Rn pSqRn´1 pSq . Rn pX q (iii) VO(X ,1) is equivalent to the vanishing order part and TC(X ,1) is equivalent to the leading Taylor coefficient part of the Birch and Swinnerton-Dyer conjecture for the Jacobian of the generic fiber X. 86 Appendix A Computational Material A.1 The motivic cycle complexes Zpnq In this section we will review the construction of Bloch’s motivic cycle complexes Zpnq “ ZpnqX for arithmetic schemes X . Simplicial structures. The standard co-simplex ∆ is the category of finite ordinal numbers rns “ t0, . . . , nu with order preserving maps as morphisms. A simplicial object A of a category C or a C-simplex is a functor A : ∆op Ñ C. We write An :“ Aprnsq and think of A as the diagram }y t ¨ ¨ ¨ A3 /// / A2 }w // / A1 y // A . 0 Simplicial objects are relevant since they may be viewed as generalizations of chain complexes. Indeed, in abelian categories these two notions are equivalent (cf. [25] Thm. 2.7). Theorem A.1. (Dold-Kan-Correspondence) Let A be an abelian category and write SimppAq for the category of A-simplices and Cě0 pAq for the category of chain complexes of A supported in non-negative degrees. There is an equivalence of categories DK : SimppAq ÝÑ Cě0 pAq, where DKpAqn “ A “ pA‚ , d‚‚ q ÞÑ DKpAq “ pDKpAq‚ , B‚ q Şn´1 i n n i“0 Ker dn and Bn “ p´1q dn : DKpAqn Ñ DKpAqn´1 . The standard co-simplex ∆ may be regarded as a full subcategory of the category of arithmetic schemes by setting Z rti |i P rnss Zrt0 , . . . , tn s ∆n “ Spec ř “ Spec řn . iPrns ti ´ 1 i“0 ti ´ 1 87 In fact, each B : rms Ñ rns gives rise to a canonical morphism of schemes ∆m Ñ ∆n induced ř by tj ÞÑ iPB´1 pjq ti . A face of ∆n is a subvariety defined by a set of equations of the kind ti1 “ ¨ ¨ ¨ “ tis “ 0. Arithmetic cycles. Let X be an arithmetic scheme of pure (Krull) dimension d. Proposition/Definition A.2. (i) For any arithmetic scheme U and any integers i, n ě 0 let ∆iU “ U ˆZ ∆i . U ‚ attains a cosimplicial structure from ∆‚ . Next, let Z n pU, iq denote the abelian group freely generated by all n-cycles of ∆iU , i.e. by all irreducible subschemes of ∆iU of codimension n that intersect all faces of ∆i properly. The proper intersection condition ensures that the inverse image of each map B : ∆iU Ñ ∆jU gives a well-defined map B ´1 : Z n pU, jq Ñ Z n pU, iq between cycle groups. We denote the corresponding simplex of abelian groups by Z n pU, ‚q. (ii) Let n ě 0. The presheaf U ÞÑ Z n pU, iq of abelian groups on Xét is already sheaf. We n p´, iq, or just Z n p´, iq if X is clear from context. We write Z n p´, ‚q denote it ZX X for the associated simplex of abelian sheaves. We define Bloch’s cycle complex Zpnq “ ZpnqX to be the chain complex of abelian sheaves on the étale site of X that arises n p´, ‚q after reindexing from the Dold-Kan correspondence applied to the simplex ZX via ‚ Ø 2n ´ ‚. More concisely, ` n ˘ ZpnqX :“ DK ZX p´, 2n ´ ‚q . ZpnqX is cohomologically concentrated in degrees ď n ` d. If d ą n it is even concentrated in degrees ď 2n. (iii) For n ă 0 one defines Zpnq :“ bn p jp,! pµp8 qr´1s where jp : X r1{ps Ñ X À denotes the canonical open embedding. (iv) For any n and m ą 0 one defines Z{mpnq :“ Zpnq{m :“ Zpnq bL Z{m, i.e. Z{mpnq is m the mapping cone of the complex Zpnq ÝÑ Zpnq. Remark A.3. The indexing in the definition of Zpnq agrees with Bloch’s use of Zpnq as well as with the use in [8]. So, we will frequently call Zpnq Bloch’s cycle complexes. Geisser [12] and Levine [18] use different indexing to obtain a cycle complex – here denoted Z̃pnq – that relates to Bloch’s complex via Z̃pnqX “ Zpd ´ nqX r2ds. Conjecture A.4. Let n ě 0. (i) (Beilinson-Soulé) ZpnqX is cohomologically concentrated in non-negative degrees. 88 (ii) If X is regular then ZpnqX is cohomologically concentrated in degrees ď n. Geisser has shown Conjecture A.4(ii) for smooth arithmetic schemes X (cf. [11] Cor. 4.2). Moreover, if n “ 0, 1, it is known for any scheme X as it is clear from Proposition A.5. (i) One has Zp0q » Zr0s and Zp1q » Gm r´1s. (ii) Let X Ñ S be regular and p invertible on S. Then, if Conjecture A.4(ii) holds true one has Z{pr pnq » µbn pr . Even without knowing the Beilinson-Soulé conjecture one may define the motivic cohomology complex RΓpX , Zpnqq using K-injective resolutions of Zpnq (as defined in [33]). A.2 GR -equivariant cohomology Write Zpnq for the GR -module p2πiqn Z. For any GR -space X let Γ˚X denote the constant sheaf functor for GR -equivariant sheaves on X. In other words, Γ˚X is given by the adjunction Γ˚X $ ΓpX, ´q. We write ZpnqX “ Γ˚X Zpnq and omit the superscript if X is clear from context. Also, write # 0 if n even n “ 1 if n odd # n “ 1 ´ n , i,n “ i´n “ 0 if i ” n mod 2 1 if i ı n mod 2 i,n “ 1 ´ i,n . For any infinite place v of F write Xv “ XF ˆF,v Spec C. Real embeddings and pairs of complex conjugate embeddings of F will be denoted by σ and tτ, τ u respectively and we let Xtτ,τ u “ Xτ > Xτ . Finally, let lpσq :“ # of connected components of Xσ pRq “ XσGR ; and lpX q :“ ÿ σ In this section we will establish the following computational Lemma A.6. One has (i) i H pGR , X pCq, Zpnqq “ (ii) $ ’ 0 ’ ’ ’ ’ rn `s ’ ’ & Z for i ă 0 for i “ 0 Zmg ‘ pZ{2qlpX qn for i “ 1 ’ ’ ’ r `s lpX q n ’ Z ‘ pZ{2q for i “ 2 ’ ’ ’ % pZ{2qlpX q for i ě 3 # H i pX8 , τ ąn Rp π˚ Zpnqq “ 0 for i ď n ` 1 pZ{2qlpX q for i ě n ` 2 lpσq. 89 i pX , Zpnqq rn `s , Zmg , Zrn `s for i “ 0, 1, 2. Moreover, (iii) One has HW 8 cotor – Z $ lpX q for n ě 0 and  ` 1 ď i ď n ` 1 ’ ’ n & pZ{2q i lpX q HW pX8 , Zpnqqtor “ pZ{2q for n ă 0 and n ` 3 ď i ď n ` 1 ’ ’ % 0 otherwise We begin with some preliminary remarks. Fix a real embedding σ of F and write X “ Xσ . Let π : X Ñ X{GR be the natural projection. Write U “ XzX GR . We have a diagram of Open-Closed-Decompositions. X GR X GR / X o i i  j U π / X{GR o j  (A.1) π U {GR where, by abuse of notation, we denote the closed and open embedding on both levels by i and j. One observes directly that the analogues of proper base change j ˚ π˚ “ π˚ j ˚ and i˚ π˚ “ π˚ i˚ “ p´qGR i˚ hold. Moreover, one has the following classical results. Proposition A.7. (cf. [13] Prop. 3.1) Let a “ apσq and l “ lpσq denote the number of connected components of U and X GR respectively1 . Then (i) U {GR is connected and X{GR has Euler characteristic 1 ´ g. (ii) 0 ď l ď g ` 1 # 2 if X{GR is orientable (iii) a “ 1 otherwise (iv) If a “ 1 then l ‰ g ` 1. If a “ 2 then l ‰ 0 and l ” g ` 1 mod 2. Moreover, each pair pa, lq satisfying the constraints (ii)-(iv) arise in the above way from some real smooth proper curve. Proof of Lemma A.6. Computation of RΓpGR , X pCq, Zpnqq. One has the decomposition RΓpGR , X pCq, Zpnqq “ à σ “ à σ RΓpGR , Xσ , Zpnqq ‘ à RΓpGR , Xtτ,τ u , Zpnqq tτ,τ u RΓpGR , Xσ , Zpnqq ‘ p2πiqn à (A.2) RΓpXtτ,τ u {GR , Zq. tτ,τ u 1 Note that a and l do in fact depend on the real embedding σ as can be seen from the elliptic curves over ? ? Qr 5s given by y 2 “ x3 ˘ 5x ´ 1. 90 The last equality follows since GR acts freely on Xtτ,τ u . Since Xtτ,τ u {GR – Xτ the cohomology of RΓpXtτ,τ u {GR , Zq is well-understood. In particular, it has no torsion. We will now analyze the first summand of (A.2). We fix a real embedding σ and adopt the notations of the comments preceding the proof. The diagram ShAb pGR , Xq π˚ ΓpX,´q / ModZrG s p´qGR 2/ AbGrps R  (A.3) ΓpX{GR ,´q ShpX{GR q commutes since F pXqGR “ pπ˚ F qpX{GR q for any GR -equivariant sheaf F on X. In particular, RΓpGR , X, ZpnqX q » RΓpX{GR , Rπ˚ ZpnqX q. We will compute the right hand side. Proposition A.7 shows that either X is the disjoint union of two copies of X{GR or π : X Ñ X{GR is the orientation cover of X{GR . Moreover, since X{GR is either non-orientable or l ą 0 one has H2 pX{GR , Zq “ 0. Also X GR “ pS 1 q>l . We analyze the restrictions of Rπ˚ Zpnq to the closed and to the open part separately. One has i˚ π˚ Γ˚X “ p´qGR i˚ Γ˚X “ p´qGR Γ˚X GR “ Hom pΓ˚X GR Z, Γ˚X GR ´q “ Γ˚X GR HomGR pZ, ´q “ Γ˚X GR p´qGR . Therefore i˚ Rπ˚ ZpnqX “ Rpi˚ π˚ Γ˚X qZpnq “ Γ˚X GR RΓpGR , Zpnqq. To compute the Galois cohomology complex we use the standard projective resolution ... 1´c/ ZpnqrGR s 1`c / ZpnqrGR s 1´c / ZpnqrGR s deg / Zpnq 0, where c P GR denotes complex conjugation. Apply HomZrGR s p´, Zpnqq. Dropping the first term yields the complex 0 2n / Z which is quasi-isomorphic to Zn ‘ / Z 2n / Z 2n / ..., À kě1 Z{2rεn ´ 2ks. In summary, we obtain i˚ Rπ˚ ZpnqX “ Γ˚X GR Zn ‘ à Γ˚X GR Z{2rεn ´ 2ks. (A.4) kě1 The restriction of Rπ˚ ZpnqX to the open part needs to be analyzed stalkwise. For x P X let Gx Ă GR denote its stabilizer and write x “ πpxq P X{GR . One has pRπ˚ Zpnqqx » 91 RΓpGx , Zpnqx q (cf. [8] Lem. 6.1). Consequently, for x P U we obtain pRπ˚ Zpnqqx » Z, proving that j ˚ Rπ˚ Zpnq » Rπ˚ ZpnqU is concentrated in degree 0. Therefore all of τ ě1 Rπ˚ Zpnq is supported on X GR . In view of (A.4) the distinguished truncation triangle for pτ ď0 , τ ě1 q equals π˚ ZpnqX ÝÑ Rπ˚ ZpnqX ÝÑ à i˚ Γ˚X GR Z{2rεn ´ 2ks ÝÑ . kě1 We apply RΓpX{GR , ´q and obtain RΓpX{GR , π˚ Zpnqq ÝÑ RΓpGR , X, Zpnqq ÝÑ à RΓpX GR , Z{2qrεn ´ 2ks ÝÑ . (A.5) kě1 We make a distinction of cases to evaluate the cohomology of the left-most complex. We will arrive at H i pX{GR , π˚ ZpnqX q i “ 0 i “ 1 i “ 2 n even n odd a“1 Z Zg a“2 Z Zg (A.6) a“1 Zg Z a“2 Zg Z First, let n be even. Then π˚ ZpnqX “ ZX{GR and the first part of table (A.6) is immediate. Now suppose n is odd. Computing stalks shows i˚ π˚ ZpnqX “ 0 and consequently π˚ ZpnqX “ j! π˚ ZpnqU . Therefore, the connecting morphisms of (A.5) must vanish. If a “ 2 then j! π˚ ZpnqU “ j! ZU {GR . The long exact sequence associated to G 0 ÝÑ j! ZU {GR ÝÑ ZX{GR ÝÑ i˚ ZX R ÝÑ 0 gives ∆ α 0 “ H 0 pX{GR , j! Zq Ñ Z ÝÑ Zl Ñ H 1 pX{GR , j! Zq Ñ Zg ÝÑ Zl Ñ H 2 pX{GR , j! Zq Ñ 0. ∆ is the diagonal embedding. H 1 pX GR , Zq{Impαq is generated by a copy of S 1 in X GR that divides X into two components since such a loop will be trivial in the fundamental group of X and hence X{GR . We infer H 1 pX{GR , j! Zq – Zg and H 2 pX{GR , j! Zq – Z. This yields the last row of (A.6). Let now n be odd and a “ 1. It follows from the classification of compact surfaces with boundary that X{GR is the l-times punctured connected sum of k “ g ` 1 ´ l many projective planes. The precise value for k follows from the equality of the two expressions for the Euler characteristic 2 ´ k ´ l “ 1 ´ g of X{GR . It is now a standard exercise to compute the Cech cohomology groups of j! π˚ ZpnqU .2 It yields the remaining row of (A.6). Now, (i) follows from the long exact sequence of cohomology associated to (A.5). 2 One may choose 2k ` 2l ` 2 “ 2pg ` 1q many simply connected open neighborhoods covering X{GR such 92 π˚ Zpnqq. The analogous computation for Tate cohoComputation of RΓpX{GR , τ ąn Rp mology is simpler since one has on stalks # À if x P X GR 0 if x P U p x , Zpnqx q “ pRp π˚ Zpnqqx “ RΓpG kPZ Z{2rεn ´ 2ks (A.7) π˚ ZpnqX “ 0 proving Consequently j ˚ Rp π˚ ZpnqX “ Rp π˚ ZpnqX “ i˚ i˚ Rp à i˚ Γ˚X GR Z{2rεn ´ 2ks (A.8) kPZ and à RΓpX{GR , τ ąn Rp π˚ Zpnqq “ RΓpX GR , Z{2qrn ´ 2ks kPZ, n ´2kă´n “ à RΓpX GR , Z{2qr´n ´ 2ks. kě1 Part (ii) follows. Computation of RΓW pX8 , Zpnqq. Recall that RΓW pX8 , Zpnqq is defined via the distinguished triangle RΓW pX8 , Zpnqq ÝÑ RΓpGR , X pCq, Zpnqq ÝÑ RΓpX pRq, τ ąn Rp π˚ Zpnqq ÝÑ . We decompose it analogously to (A.2) and – together with (A.5) – we get for n ě 0 n`n 2 RΓpX{GR , π˚ Zpnqq ÝÑ RΓW pX{GR , Zpnqq ÝÑ à RΓpX GR , Z{2qrn ´ 2ks ÝÑ k“1 The long exact sequence on cohomology together with (A.6) proves the first case of (iii). For n ă 0 we similarly get the distinguished triangle n ´ n` 2 à RΓpX GR , Z{2qr´n ´ 2k ´ 1s ÝÑ RΓW pX{GR , Zpnqq ÝÑ RΓpX{GR , π˚ Zpnqq ÝÑ k“1 The remaining cases of (iii) follow. l The analogous results for SpCq are much simpler to prove. We write iR : SpRq ãÑ S8 and iC : S8 zSpRq ãÑ S8 for the closed immersions of the collection of all real points and pairs of complex conjugate points respectively. Lemma A.8. One has that no four of them have common non-trivial intersection and such that each boundary of X{GR intersects precisely two open neighborhoods. The resulting Cech complex has three terms and one verifies by direct computation that each of its its cohomology groups is torsion-free. We leave further details to the reader. 93 (i) $ ’ ’ & 0 i Zrn `s ’ ’ % pZ{2qri,n for i “ 0 # 0 for i ď n ` 1 pZ{2qri,n for i ě n ` 2 H pGR , SpCq, Zpnqq “ (ii) i H pS8 , τ ąn for i ă 0 π˚ Zpnqq “ Rp for i ě 1 i pS , Zpnqq rn `s , 0 for i “ 0, i ‰ 0. Moreover, (iii) One has HW 8 cotor – Z i HW pS8 , Zpnqqtor “ $ r ’ ’ & pZ{2q for n ě 0 and n ` 1 ď i ă n ` 1, i ” n mod 2 pZ{2qr for n ă 0 and n ` 3 ď i ă n ` 1, i ı n mod 2 ’ ’ % 0 otherwise Proof. We follow the proof of Lemma A.6. As Sσ “ SF ˆ Spec C “ Spec C is just a point F,σ the analogue of (A.1) collapses to the identity of a one-point space. When combining the analogues of (A.2) and (A.4) we immediately arrive at ˜ SpCq Rπ˚ Zpnq “ piC q˚ Γ˚S8 zSpRq Z ‘ piR q˚ ¸ Γ˚SpRq Zn ‘ Γ˚SpRq à Z{2rn ´ 2ks . (A.9) kě1 Applying RΓpS8 , ´q yields RΓpGR , SpCq, Zpnqq “ Zrn `s ‘ à RΓpSpRq, Z{2qrn ´ 2ks. kě1 Part (i) follows. Next, mimicking the computations (A.7) and (A.8) yields π˚ ZpnqSpCq » piR q˚ Γ˚SpRq Rp à Z{2rn ´ 2ks. (A.10) kPZ (ii) follows after truncating and applying RΓpS8 , ´q. For (iii) consider the distinguished triangle defining RΓW pS8 , Zpnqq. For n ě 0 one gets rn `s RΓW pSi , Zpnqq » Z n`n 2 ‘ à RΓpSpRq, Z{2qrn ´ 2ks. k“1 Similarly, for n ă 0 one has n ´ n` 2 à RΓpSpRq, Z{2qr´n´2k´1s ÝÑ RΓW pS8 , Zpnqq ÝÑ Zrn `s ÝÑ k“1 proving the final part of the claim. 94 1 pS , Zpnqq mg , 0 for i “ Corollary A.9. Write 1lpX q “ lpX q ´ r. One has 1HW 8 cotor – Z 1, i ‰ 1. Moreover, 1 i HW pX8 , Zpnqqtor “ $ 1 lpX q ’ ’ & pZ{2q for n ě 0 and n ` 1 ď i ď n ` 1 1 pZ{2q ’ ’ % 0 lpX q for n ă 0 and n ` 3 ď i ď n ` 1 otherwise In particular, χ ¯p´1qi `p 1 ˘ ź´ i,n #Tor 1HW,8 pX q “ 1. R ΓW pX8 , Zpnqq “ iPZ Corollary A.10. For all integers n one has ¯p´1qi ź´ n n`n i,n “ 2p´1q 2 r . χpRΓW pS8 , Zpnqqq “ #Tor HW,8 pSq iPZ A.3 Comparison between motivic and completed motivic cohomology Recall the triangle (2.26). Building on the work done in the last section we will show that the term τ ąn Rp π˚ ZpnqX pCq controlling the discrepancy between motivic and completed motivic cohomology allows a decomposition into motivic degrees analogous to Theorem 2.11. Lemma A.11. Let π8 : X8 Ñ S8 be the structure map of X8 . Suppose π8 has a section s8 : S8 Ñ X8 , or, equivalently, lpσq ą 0 for every real place σ. Then – in the derived category of abelian sheaves on S8 – the complexes Rp π˚ ZpnqSpCq and Rp π˚ ZpnqSpCq r´1s split p off as direct summands of Rπ8,˚ Rp π˚ ZpnqX pCq .3 When writing R1 π8,˚ Rp π˚ ZpnqX pCq for the remaining summand we arrive at the canonical decomposition p π˚ ZpnqX pCq » Rp π˚ ZpnqSpCq ‘ R1 π8,˚ Rp Rπ8,˚ Rp π˚ ZpnqX pCq r´1s ‘ Rp π˚ Zpn´1qSpCq r´2s or, equivalently, π˚ Zpnq » piR q˚ Rπ8,˚ Rp à p π˚ Zpnqr´1s ‘ piR q˚ Z{2SpRq rn ´2ks ‘ R1 π8,˚ Rp kPZ à Z{2SpRq rn ´2k´1s kPZ Proof. The equivalence of the two decompositions is (A.10); we will prove the latter. We adopt the notations of the proof of Lemma A.6. We will also need the restrictions πR “ π8 |X pRq and sR “ s8 |SpRq . By virtue of (A.8) one has à à à Rπ8,˚ Rp π˚ Zpnq “ Rπ8,˚ i˚ Γ˚X GR Z{2 “ Rπ8,˚ i˚ Z{2SpRq “ piR q˚ RπR,˚ Z{2SpRq . kPZ 3 kPZ kPZ Beware the two different roles of the letter π. π8 is derived from the structure map X pCq Ñ SpCq p˚ is the Tate modification of the morphism of topoi ShpGR , X pCqq Ñ ShpX8 q induced by the natural while π projection X pCq Ñ X8 . 95 So, it suffices to show that Z{2SpRq and Z{2SpRq r´1s split off as direct summands of RπR,˚ Z{2SpRq . The category of constant sheaves on the finite point space SpRq is equivalent to the category of abelian groups via the global sections functor. Therefore we have to prove a decomposition of the form p RΓpX pRq, Z{2q » RΓpSpRq, Z{2q ‘ R1 ΓpX pRq, Z{2q ‘ RΓpSpRq, Z{2qr´1s. Using the derived functor formalism as in the proof of Theorem 2.8 the identity πR sR “ id shows immediately that RΓpSpRq, Z{2q splits off as direct summand of RΓpX pRq, Z{2q (see also Remark 2.9). Moreover, Poincaré duality gives RΓpX pRq, Z{2q_ » RΓpX pRq, Z{2qr1s and RΓpSpRq, Z{2q_ » RΓpSpRq, Z{2q. Therefore RΓpSpRq, Z{2qr´1s » RΓpSpRq, Z{2qr1s_ splits off of RΓpX pRq, Z{2q as well. RΓpSpRq, Z{2q and RΓpSpRq, Z{2qr´1s must in fact be distinct direct summands for degree reasons. π˚ ZpnqX pCq . Therefore, the decompoEvidently the same proof holds for any truncation of Rp sition of Theorem 2.11 extends to completed motivic cohomology. Corollary A.12. Suppose π : X Ñ S has a section s : S Ñ X satisfying FPB(s,n) for all integers n ě 2. Then the complexes ZpnqS and Zpn ´ 1qS r´2s split off as direct summands of Rπ˚ ZpnqX in the derived category of sheaves on the Artin-Verdier étale site X “ X ét of p X . When writing R1 π˚ ZpnqX r´1s for the remaining summand we arrive at the canonical decomposition p Rπ˚ ZpnqX » ZpnqS ‘ R1 π˚ ZpnqX r´1s ‘ Zpn ´ 1qS r´2s. (A.11) Remark A.13. The decomposition (2.14) of Theorem 2.11 cannot generally hold without assuming the existence of a section. Indeed, if the generic fiber X “ XF does not have an F -rational point then lpX q “ 0 and consequently H i,n pX q “ H i,n pX q for all i, n. However, we will see below that H i,n pX q “ H i,n pSq “ 0 for i ě 7, n ě 1 while at the same time H i,n pSq “ pZ{2qri,n for i ě 5, n ě 1. If a section exists, then lpX q can be understood in terms of the Jacobian J “ Jac XF of the generic fiber. In fact, one has Proposition A.14. Let X be a smooth proper real algebraic curve. Write J “ Jac XpCq and let l be the number of connected components of XpRq. If l ą 0 then H 1 pGR , Jq – pZ{2ql´1 . Proof. Combine [13] Prop. 1.1, Prop. 1.3, and Prop. 3.2(2). 96 Computation of torsion parts. We will conclude with a summary of all information on torsion for motivic and completed motivic cohomology. Recall the notations T?i,n “ Tor H i,n p?qcodiv for ? “ S, S, X , X and 1T?i,n “ Tor 1H i,n p?qcodiv for ? “ X , X . Proposition A.15. Suppose π : X Ñ S has a section s : S Ñ X satisfying FPB(s,n) for all integers n ě 2. Write 1lpX q “ lpX q ´ r. Then TSi,n , TSi,n and 1TXi,n , 1TXi,n are given as in the tables below. TSi,n iď0 i“1 TSi,n „ ClF „ µF pZ{2qri 0 0 ClF µF 0 0 µF ClF Br O pZ{2qri 0 µF ClF 0 0 i ď 0 i “ 1 i “ 2 3ďiďn`1 n`2 ď i TSi,n 0 TS1,n TS2,n pZ{2qri,n pZ{2qri,n 0 TS1,n TS2,n pZ{2qri,n 0 ną1 TSi,n i ď n ` 2 n`3 ď i ď 1 1 i,n TX 0 0 pZ{2qri,n iď1 i“2 i“2 i“3 4ďi „ TS2,1´n „ TS1,1´n pZ{2qri,n 0 nă0 1 i,n TX 4ďi 0 n“1 TSi,n i“3 0 n“0 TSi,n i“2 TS2,1´n i“3 TS1,1´n 0 i“4 iě5 0 „ 1TX4,2´n „ 1TX3,2´n „ 1TX2,2´n pZ{2q lpX q 0 1 4,2´n TX 1 3,2´n TX 1 2,2´n TX 0 0 Tor Pic0 X ClF Br X Br O Pic X „ TorCl F pZ{2q lpX q 0 Tor Pic0 X ClF XpX{F q Tor Pic0 X ClF 0 0 1 2,n TX 1 2,n TX 1 3,n TX 1 3,n TX „ 1TX4,n pZ{2q lpX q 1 4,n TX 0 n“0 n“1 n“2 0 0 1 1 1 97 1 i,n TX 1 i,n TX i ď 1 i “ 2, 3, 4 5 ď i ď n`1 n ` 2 ď i 0 ną2 0 1 i,n TX 1 i,n TX 1 i,n TX 1 i,n TX pZ{2q lpX q 1 0 pZ{2q lpX q iďn`2 n`3 ď i ď 1 0 0 0 pZ{2q lpX q nă0 1 pZ{2q lpX q i “ 2, 3, 4 1 5ďi „ 1TX6´i,2´n pZ{2q lpX q 1 1 1 6´i,2´n TX 0 p In particular, RΓpS, Zpnqq and R1 ΓpX , Zpnqq are perfect complexes. Proof. The vanishing of 1TXi,n and 1TSi,n for i ď 1 and i ď 0 respectively has been established in Proposition 2.20. The torsion groups for n “ 1 and near-central i have already been computed in in the proof of Proposition 2.17. The remaining entries are easily obtained from Zp0q » Z, Zp1q » Gm r´1s and Artin-Verdier duality. Throughout, use the triangle (2.26) to translate between motivic and completed motivic cohomology. Corollary A.16. One has (i) $ ’ 2,1´n ’ ’ n n`n r #T ’ p´1q S ’ 2 ’ 2 ź` ˘p´1qi & #TS1,1´n i,n #Tor H pSq “ ’ 2,n ’ iPZ ’ n ´1 r #TS ’ p´1qn p n` q ’ 2 ’ 2 % #TS1,n (ii) ¯p´1qi ź´ #Tor 1H i,n pX q “ iPZ A.4 $ ’ ’ ’ ’ ’ ’ & 1 #XpX{F q Tor Pic0 X # ClF ˆ ’ ’ # 1TX2,n ¨ # 1TX4,n ’ ’ ´n 1lpX q ’ ’ % 2 # 1TX3,n for n ď 0 for n ě 1 ˙2 for n “ 1 for n ‰ 1 Supplementary material for derived de Rham cohomology In this section π : X Ñ S will denote any map of schemes. 98 Construction of the cotangent complex LX{S . An adjunction T $ U between categories A, B – which we write as A o / T U B – gives rise to a simplicial structure pT, U q‚ on the endofunctor category End B as follows. The adjunction data may be thought of as a pair of natural transformations α : idA ÝÑ U T and β : T U ÝÑ idB that satisfy the compatibility relations pβ ˚ T q ˝ pT ˚ αq “ idT pU ˚ βq ˝ pα ˚ U q “ idU . and We define pT, U qn “ pT U qn`1 and let the (co)boundary maps be given by the natural transformations pT U qi ˚ β ˚ pT U qn´i : pT U qn`1 ÝÑ pT U qn , pT U qi´1 T ˚ α ˚ U pT U qn´i : pT U qn ÝÑ pT U qn`1 . Repeated application of β gives a canonical map pT, U q‚ Ñ idB , i.e. a unique compatible collection of natural transformations pT, U qn Ñ idB . Now, fix a scheme X and let ShpXZar q and ShRings pXZar q denote the topoi of sheaves of sets and rings on the Zariski site XZar of X. We fix O P ShRings pXZar q and let ShO-Alg pXZar q denote the topos of O-algebras on XZar . We will apply the previous construction to the adjunction ShpXZar q o Or´s / Ou ShO-Alg pXZar q , where the functor Or´s “ SymO Op´q assigns to any sheaf of sets F on XZar the sheaf of O-algebras OrFs which is the sheafification of the presheaf that assigns to each Zariski-open U Ă X the OpU q-algebra freely generated by FpU q regarded as a set. Let PO p´q denote the resulting End ShO-Alg pXZar q-simplex. If X comes with a structure map π : X Ñ S the above construction may be carried out for O “ π ´1 OS . Applying the resulting functors to the π ´1 OS -module OX yields the X{S Shπ´1 OS -Alg pXZar q-simplex P‚ X{S “ Pπ´1 OS pOX q. Each Pi has an algebraic de Rham reso- X{S lution Pi Ñ Ω‚ X{S ´1 P {π O simplices “ Ω‚ X{S ´1 . So, we obtain a complex of Shπ´1 OS -Alg pXZar qPi {π OS S i O O /// /// Ω2 X{S // Ω2 // /// ΩP X{S O // Ω X{S P0 O /// P X{S // P X{S / Ω2P X{S 2 O Ω‚ X{S “ P‚ O / ΩP X{S 2 O // P X{S 2 O P1 1 1 X{S O P0 0 99 X{S The rows Ωk X{S of Ω‚ X{S may be regarded as P‚ P‚ X{S P‚ P‚ -modules and we may use the morphism Ñ OX to define L̃kX{S :“ Ωk X{S b OX P‚ X{S P‚ as a simplicial object of ShOX -Mod pXZar q. Write LkX{S “ DKpL̃kX{S q for the complex of OX -modules associated to L̃kX{S via the DoldKan correspondence. LX{S :“ L1X{S is the cotangent complex of π : X Ñ S. One has Ź LkX{S “ k LX{S . We write ˜ ¸ ż ż LΩ‚X{S “ L‚X{S “ DK Ω‚ X{S b OX P‚ X{S P‚ for the totalization of the double complex arising from Ω‚ X{S b OX after applying the P‚ Dold-Kan correspondence to its rows. Moreover, we define a filtration ˜ ¸ ż ż F m :“ Film LΩ‚X{S :“ Lěm DK Ωěm b OX X{S X{S “ P‚ X{S P‚ m ‚ m :“ LΩ‚ ‚ ‚ m m`1 . m and write LΩďm X{S {Fil LΩX{S as well as gr LΩX{S “ F {F X{S :“ LΩX{S {F The de Rham conductor ApX q. The de Rham or Kato conductor of π 1 : X Ñ Z ź` ˘p´1qi ApX q :“ #Hi pX , Ω‚X ,tors q P Qˆ iPZ encapsulates the discrepancy between algebraic de Rham and derived de Rham cohomology (cf. Proposition 2.23(iii)). Since Ω‚X ,tors is concentrated on the non-smooth points of X the conductor ApX q depends only on the bad fibers of X . In many instances it can be computed explicitly. We give one example. Proposition A.17. Let π 1 : X Ñ Z have semistable reduction at every prime p. Write ip : Zp ãÑ X for the closed embedding of the subscheme of singular points Zp of the special fiber Xp into X . Then ΩX {Z,tors “ 0 Ω2X {Z “ and à p pip q˚ Z{p. In particular, ApX q “ ź p#Zp . p Proof. First, observe that Z “ Ť p Zp Ă X must be a finite collection of closed points. Let x be a singular closed point of X and let x ãÑ X be a corresponding geometric point. Semistability means that ˆ sh OX ,x “ OX ,x – Zru, vs uv ´ p ˙sh , pu,v,pq 100 i.e. étale locally at x the differentials of X coincide with the differentials of the scheme X0 “ Spec Zru,vs uv´p over Z at its singular point O “ pu, v, pq. So, when writing f pu, vq “ uv ´ p, the claim follows from the direct computations ˆ ` ˘ ΩX0 {Z O,tors “ ΓpX0 , ΩX0 {Z qtors “ OX ,x du ‘ OX ,x dv df ˙ “0 tors and ¯ ´ OX ,x ¯ du ^ dv – Z{p. “ ΓpX0 , Ω2X0 {Z q “ ´ Ω2X0 {Z Bf Bf O , Bu Bv This exercise may be repeated for singularities described by different equations f . A general formula for ApX q — which involves Swan characters of Galois representations given by the l-adic étale cohomology of XQ — has been found by Bloch in [3], Prop. 1.1. A.5 Overview of computed cohomology groups The following tables summarize the results for the ranks of the cohomology groups associated to X as computed in Chapter 2 and pair them with the corresponding cohomology groups for S (see [8] Sec. 5.8 for their derivations). As indicated before, we observe in all cases a decomposition pattern as it is implied by decompositions of the kind H?i,n pX q “ H?i,n pSq ‘ 1H?i,n pX q ‘ H?i´2,n´1 pSq. We also provide torsion information for motivic and Weil-étale motivic cohomology groups. A.5.1 Rank tables Ranks of Deligne cohomology. i,n dimR HD pSq i,n dimR HD pX q nď0 n“1 ně2 i“0 i“1 i“2 mg n r ` s i“3 n r ` s n r ` s (A.12) r`s r`s r`s n r ` s n r ` s mg n r ` s 101 i pX , Zpnqq. Ranks of HW 8 i,n dimR HW,8 pSq i“0 i,n dimR HW,8 pX q i“1 i“2 (A.13) n r ` s any n n r ` s mg n r ` s Ranks of de Rham cohomology. i,n dimR HdR pSq i“0 i“1 i“2 i,n dimR HdR pX q n“1 ně2 H ‚ pFC , CqGR H ‚ pX pCq, O X pCq m qGR m H ‚ pFC , CqGR m H ‚ pX pCq, CqGR m (A.14) mg 2mg m i“2 i“3 i“4 mg n r ` s mg s Ranks of compact support cohomology. dimR Hci,n pSq dimR Hci,n pX q nă0 n“0 n“1 n“2 i“1 n r ` s n r ` s r`s´1 r`s´1 (A.15) 1 rk Pic X r`s´1 1 102 Ranks of Weil-étale cohomology with compact support. i,n rk HW,c pSq i,n rk HW,c pX q nă0 n“0 n“1 n“2 ną2 A.5.2 i“1 i“2 n r ` s n r ` s n r ` s n r ` s r`s´1 r`s´1 r`s´1 mg ` r ` s ´ 1 m´1 m´1 i“3 i“4 n r ` s n r ` s mg ` s s i“5 (A.16) 1 rk Pic X ` m ´ 1 rk Pic X `r`s´1 r ` s ´ 1 m m 2mg m´1 2mg m m m Motivic and Weil-étale motivic cohomology tables The entries of the following tables are valid up to 2-torsion. 1 ně3 n“2 n“1 n“0 nă0 i,n HW pX q i,n HW pSq i“4 TS1,1´n Zrn `s ‘ TS2,1´n TS2,n´1 ‘ 1TX4,n i“4 pOFˆ q_ pQ{Zqs ‘ TS1,2 i“5 Q{Z i“6 Pic X TS2,2 Zmg ‘ TS2,2 ‘ 1TX2,2 TS2,n Zmg ‘ TS2,n ‘ 1TX2,n OFˆ Zs ‘ TS1,2 Zs ‘ TS1,2 Zn r`s ‘ TS1,n Zn r`s ‘ TS1,n Z ClF ClF ‘ 1TX4,2 ‘ pOFˆ q˚ OFˆ ClF ‘ pOFˆ q˚ F ‘ 1 3,2 TX Zn r`s ‘ TS1,n´1 ‘ 1TX3,n OFˆ ‘ 1TX3,2 Br X ‘ pPic X q˚ Z Zmg ‘ µ µF TS2,n´1 ‘ 1TX4,n ClF ‘ 1TX4,2 Tor Pic X ‘ pOFˆ q˚ Zs ‘ TS2,2 ‘ 1TX2,2 Z µF TS1,2 Zrn `s ‘ TS2,1´n ‘ 1TX4,2´n Zmg ‘ TS1,1´n ‘ 1TX3,2´n Zn r`s ‘ TS2,2´n ‘ 1TX2,2´n TS1,2´n i“3 i“2 Z i“1 Zn r`s ‘ TS1,n´1 ‘ 1TX3,n ClF ‘ 1TX4,2 TS2,2 Zmg ‘ TS2,2 ‘ 1TX2,2 TS2,n Zmg ‘ TS2,n ‘ 1TX2,n OFˆ ‘ 1TX3,2 pPic X q_ Pic X Zs ‘ TS1,2 Zs ‘ TS1,2  Z n r`s ‘ TS1,n Zn r`s ‘ TS1,n Br X Q{Z pQ{Zqmg ‘ TS2,2 ‘ 1TX2,2 ClF pOFˆ q_ ‘ 1TX3,2 ClF ‘ 1TX4,2 Z OFˆ OFˆ pOFˆ q_ i“0 i“5 pQ{Zqn r`s ‘ TS1,1´n ‘ 1TX3,2´n pQ{Zqmg ‘ TS2,2´n ‘ 1TX2,2´n pQ{Zqn r`s ‘ TS1,2´n Q{Zrn `s ‘ TS1,1´n TS2,1´n TS2,1´n ‘ 1TX4,2´n i“3 i“2 ClF i“1 Z i“0 Weil-étale motivic cohomology. ně3 n“2 n“1 n“0 nă0 H i,n pX q H i,n pSq Motivic cohomology. (A.18) (A.17) 103 A.6 de Rham cohomology coming from derived integral structure ζ ˚ pX , nq Taylor coefficient encodes leading conjecturally line ∆pX {Z, nq; is fundamental this triangle associated to Z-determinant ‚,n pX q of Har,c characteristic equal to Euler conjecturally of ζpX , sq at n Vanishing order   i´1,n HdR pX qR  i,n HW,c pX qR o i,n Har,c pX q o  [ D   / H i,n pX q R R ´b1 ´b1 H i,n pX qR o mapping fiber mapping fiber add one copy of Z for each Q{Z-summand of H i`1,n pX q; drop all Q{Z-summands distinguished triangle induced by n n Ω‚ X pCq {F r´1s Ñ RpnqD Ñ p2πiq R Ñ motivic cohomology; directly computable for n “ 0, 1; W # / H i,n pX q finitely generated groups; defined via the distinguished triangle /  isomorphic to Betti cohomology H i pX pCq, RpnqqGR i,n HW,8 pX qR R ´b1 i,n HW,8 pX q ' ( / RHompRΓpX , Qp2´nqq, Qp´6qq Ñ RΓpX , Zpnqq Ñ RΓW pX , Zpnqq Ñ H i,n pX q for remaining twists n use Artin-Verdier duality and BpX , nq equal up to 2-torsion H i,n pX q completed motivic cohomology i,n HW,c pX q computable via conjectural perfect pairing Hci,n pX q BpX , nq : Hci,n pX q ˆ H 4´i,2´n pX qR Ñ R Beilinson regulator map R ´b1 combine degrees i and i´1 Diagram of cohomology groups 104 2´n C 2´n “ tcvj uvj 3,n HW pX qR for n ď 0 01 “ tη u BdR vj vj H 0 pX , ωX {Z qR Hc2,n pX q for n ě 2; see Section 3.2 for definition via 1Hc2,1 pX q – 1H 2,1 pX qR 1,n pX q via 1Hc2,n pX q – HD P B `;n for n “ 1 for n ď 0 1 pX {Rq – H 1 pX , O q ‘ H 0 pX , ω via HdR 8 X R X {Z qR n BddR 1 pX {Rq 10 Y B 01 H 1 pX pCq, CqGR “ HdR BdR “ BdR 8 dR 1 10 Serre dual of BdR 3,n via 1HW pX qR – HompH 2,2´n pX q, Rq 2,n via 1HW pX qR – H 2,n pX qR B n “ B `;n Y B `;n´1 for any n; via H 1 pX pCq, CqGR “ H 1 pX pCq, RpnqqGR ‘ H 1 pX pCq, Rpn´1qqGR 10 “ tω u BdR vj vj H 1 pX , OX qR 1 1 0 P basis of pPic0 XqR ; Dp basis of pΛp qR p Dp Ť 0 P1 “ P Y Cn R 2,n HW pX qR for n ě 2 H 2,1 pX q R “ nu C n “ tcvj vj H 2,n pX qR for n ě 2 1,n via HD pX q – H 1 pX pCq, RpnqqGR X Pic X generates image of Pic ClF inside p ClF qR B `;n 1,n HD pX q for n ď 0 2,n pX q – H 1 pX pCq, Rpn ´ 1qqGR via HD 1,n pX qR – H 1 pX pCq, RpnqqGR via HW,8 “ p2πiqn B ˘ Poincaré dual of B ` comments P B `;n´1 2,n pX q for n ě 2 HD ¯ B `;n 1,n pX qR for all n HW,8 Pic0 X ClF B `;n H 1 pX pCq, RpnqqGR for all n ´ ´ B ´ “ tδvj uvj H 1 pX pCq, Rq´ 1 ` B ` “ tδvj uvj H 1 pX pCq, Rq` basis of integral structure Overview of integral structures occurring in ∆pX {Z, nq Cohomology group A.7 105 106 Appendix B Bibliography [1] H. 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