The Conjecture of Birch and Swinnerton-Dyer for Elliptic Curves with Complex Multiplication by a Nonmaximal Order Citation Colwell, Jason Andrew (2004) The Conjecture of Birch and Swinnerton-Dyer for Elliptic Curves with Complex Multiplication by a Nonmaximal Order. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/G40X-ST27. https://resolver.caltech.edu/CaltechETD:etd-04012004-151307 Abstract The Conjecture of Birch and Swinnerton-Dyer relates an analytic invariant of an elliptic curve -- the value of the L-function, to an algebraic invariant of the curve -- the order of the Tate--Shafarevich group. Gross has refined the Birch--Swinnerton-Dyer Conjecture in the case of an elliptic curve with complex multiplication by the full ring of integers in a quadratic imaginary field. It is this version which interests us here. Gross' Conjecture has been reformulated, by Fontaine and Perrin-Riou, in the language of derived categories and determinants of perfect complexes. Burns and Flach then realized that this immediately leads to a refined conjecture for elliptic curves with complex multiplication by a nonmaximal order. The conjecture is now expressed as a statement concerning a generator of the image of a map of 1-dimensional modules. We prove this conjecture of Burns and Flach. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: complex multiplication; elliptic curve; equivariant Tamagawa number conjecture; L-function; Tate-Shafarevich group Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Flach, Matthias Thesis Committee: Flach, Matthias (chair) Ramakrishnan, Dinakar Goins, Edray Aschbacher, Michael Defense Date: 18 November 2003 Record Number: CaltechETD:etd-04012004-151307 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-04012004-151307 DOI: 10.7907/G40X-ST27 ORCID: Author ORCID Colwell, Jason Andrew 0000-0002-1041-4739 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1239 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 02 Apr 2004 Last Modified: 06 Jan 2021 00:52 Thesis Files Preview PDF (thesis.pdf) - Final Version See Usage Policy. 463kB Preview PDF (abstract.pdf) - Final Version See Usage Policy. 24kB Repository Staff Only: item control page

Jason Colwell Thesis Abstract November 2003 The Conjecture of Birch and Swinnerton-Dyer relates an analytic invariant of an elliptic curve – the value of the L-function, to an algebraic invariant of the curve – the order of the Tate–Šafarevič group. Gross has refined the Birch–Swinnerton-Dyer Conjecture in the case where the endomorphism ring O is the full ring of integers OK . It is this version which interests us here. Gross’ Conjecture has been reformulated, by Fontaine and Perrin-Riou, in the language of derived categories and determinants of perfect complexes. Burns and Flach then realized that this immediately leads to a refined conjecture for elliptic curves with complex multiplication by a nonmaximal order. The conjecture is now expressed as a statement concerning a generator of the image of a map of 1-dimensional modules. We prove this conjecture of Burns and Flach. 1  The Conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication by a nonmaximal order. Thesis by Jason Colwell In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2004 (Defended 18 November 2003) ii c 2004 Jason Colwell All Rights Reserved iii Acknowledgements First, I would like to thank my advisor, Matthias Flach, for his instruction during my graduate studies, including his suggestion of this topic, and his help in the writing of this thesis. I am grateful to the Department of Mathematics at Caltech for providing a congenial environment for study and research. I would like to thank my father and mother for their support and encouragement throughout my doctoral program. Finally, I would like to thank my girlfriend, Charlene, for her constant encouragement and personal support during the writing of this thesis. iv Abstract The Conjecture of Birch and Swinnerton-Dyer relates an analytic invariant of an elliptic curve – the value of the L-function, to an algebraic invariant of the curve – the order of the Tate–Šafarevič group. Gross has refined the Birch–Swinnerton-Dyer Conjecture in the case of an elliptic curve with complex multiplication by the full ring of integers in a quadratic imaginary field. It is this version which interests us here. Gross’ Conjecture has been reformulated, by Fontaine and Perrin-Riou, in the language of derived categories and determinants of perfect complexes. Burns and Flach then realized that this immediately leads to a refined conjecture for elliptic curves with complex multiplication by a nonmaximal order. The conjecture is now expressed as a statement concerning a generator of the image of a map of 1-dimensional modules. We prove this conjecture of Burns and Flach. v Contents Acknowledgements iii Abstract iv 0 Introduction. 1 1 The conjecture of Birch and Swinnerton-Dyer. 5 2 The Weil restriction. 9 3 The conjecture of Gross. 12 3.1 Choice of bases. . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 The conjecture. . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4 The language of perfect complexes. 18 4.1 Determinants of perfect complexes. . . . . . . . . . . . . . . . 18 4.2 The setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Four vector spaces. . . . . . . . . . . . . . . . . . . . . . . . . 22 4.4 Galois cohomology. . . . . . . . . . . . . . . . . . . . . . . . . 23 4.5 Reformulation of Gross’ Conjecture. . . . . . . . . . . . . . . . 29 5 Elliptic curves with nonmaximal endomorphism ring. 47 vi 5.1 The Burns–Flach Conjecture. . . . . . . . . . . . . . . . . . . 52 5.2 A refinement in the case F (Etors )/K is abelian. . . . . . . . . 53 6 Translation into the terminology of Kato. 59 7 Proof of the conjecture. 68 7.1 A universal situation. . . . . . . . . . . . . . . . . . . . . . . . 68 7.2 The strategy of proof. 74 . . . . . . . . . . . . . . . . . . . . . . 8 A Λ∞ -basis of ∆Λ∞ (OK,S , F∞ ). 76 9 The image of the basis. 86 10 Examples. 92 10.1 Elliptic curves with rank 0. . . . . . . . . . . . . . . . . . . . . 93 10.2 Elliptic curves with complex multiplication by a nonmaximal order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 10.3 A first attempt. . . . . . . . . . . . . . . . . . . . . . . . . . . 99 10.4 A second search. . . . . . . . . . . . . . . . . . . . . . . . . . 103 10.5 The code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Bibliography 119 1 Chapter 0 Introduction. The Conjecture of Birch and Swinnerton-Dyer relates an analytic invariant of an elliptic curve – the value of the L-function, to an algebraic invariant of the curve – the order of the Tate–Šafarevič group. In our situation, the elliptic curve E is defined over a number field F . The L-function, like the classical Riemann zeta function, is defined as a product over prime ideals of F . The Tate–Shavarevich group is the group of 1cohomology classes of E which are locally trivial at each place of F . For most elliptic curves E, the ring of endomorphisms End(E) is isomorphic to Z. However, there are some elliptic curves whose endomorphism ring is larger. In this case, E is said to have complex multiplication. It is elliptic curves with this additional structure which interest us here. The basic theory of the subject shows that if E has complex multiplication, then End(E) is isomorphic to an order O in a quadratic imaginary field K. Gross has refined the Birch–Swinnerton-Dyer Conjecture in the case where O is the full ring of integers OK . It is this version which interests us here. He uses the fact that the L-function of an elliptic curve with complex mul- 2 tiplication agrees with the product of two L-functions of a Grössencharacter of F , associated to E, with values in K. Gross’ idea is to consider the two L-functions separately, making a conjecture about each, viewing it as the Kequivariant L-function. He formulates a complex regulator in place of the real regulator used in the original conjecture. As well, he uses the order ideal of a group in place of the cardinality of the group. Rubin has proved the p-part of Gross’ conjecture in the case where F = K (i.e., the class number of K is 1) and p does not divide |µOK | (the number of roots of unity in K), under the condition rank(E(F )) = 0. Some cases of Gross conjecture with F 6= K will follow from our results below. Gross’ Conjecture has been reformulated, by Fontaine and Perrin-Riou, in the language of derived categories and determinants of perfect complexes. Burns and Flach then realized that this immediately leads to a conjecture which is a refinement in the case where O 6= OK . The conjecture is now expressed as a statement concerning a generator of the image of a map of 1dimensional modules. We wish to prove the p-part of this conjecture of Burns and Flach for elliptic curves with complex multiplication by a nonmaximal order, i.e., where O 6= OK , and where necessarily F 6= K. We shall use Rubin’s restrictions rank(E(F )) = 0 F (Etors )/K is abelian p 6 | |µOH | 3 (H being the Hilbert Class field of K). In fact we shall also assume for simplicity that K has class number 1 (i.e., H = K). The formulation of Burns–Flach naturally allows a further refinement of the conjecture, dealing with the Weil restriction B of E, and considering Rp modules in place of Op -modules, where R = End(B). (The conjecture will be over the endomorphism algebra R of B, which is much larger than K.) We shall prove this conjecture by reducing it to the Main Conjecture for imaginary quadratic fields. At this point we need to make the further assumption that p 6 | [K f0 p : K], where f0 is the prime-to-p part of the conductor of B and K m /K denotes the ray class field of conductor m, because Rubin’s proof of the Main Conjecture requires this condition. We shall use a method of Kato to address this situation, first dualizing the modules involved, then passing to a situation universal with respect to invertible sheaves over p-adic rings. The conjecture is then seen to follow from two known results, the Explicit Reciprocity Law and Rubin’s Main Conjecture. We first discuss the conjectures which are ancestors of the Burns–Flach Conjecture under consideration, and the translation between the language of the original Birch–Swinnerton-Dyer Conjecture and that of perfect complexes. Next, a deduction of the Burns–Flach Conjecture, from the Main Conjecture and the Explicit Reciprocity Law, is given. Finally, we discuss numerical examples of the conjectures. 4 5 Chapter 1 The conjecture of Birch and Swinnerton-Dyer. We will give the Birch–Swinnerton-Dyer Conjecture in its original generality, applying not only to elliptic curves but to abelian varieties in general. We follow [9] in our exposition. Let A be an abelian variety of dimension g which is defined over a number field F . Write L(A/F, s) for the L-series of A over F , which is defined by the Euler product L(A/F, s) = Y Pw (A/F, Nw−s )−1 , w6 |∞ where for any finite place w of F Pw (A/F, x) := det(1 − xφw | (T` (A) ⊗ Q` )Iw ), the characteristic polynomial of the action of Frobenius (φw ) upon the inertia `-adic Tate quotient module (T` (A) ⊗ Q` )Iw (which is the co-invariants, under the action of the inertia group Iw , of the `-adic Tate module T` (A) ⊗ Q` ). 6 This polynomial is independent of ` and has coefficients in Z. The L-series is convergent in the half-plane {s ∈ C : Re(s) > 32 }. Let III(A) be the Tate-Šafarevič group ! ker H 1 (F, A) → Y H 1 (Fw , A) w of A over F . (The product is taken over all places w of F .) The following is conjectured to be true, and will be assumed here. Assumption 1.0.1. III(A) is finite. A constant is defined using the Néron model, the invariant differentials on A and their period integrals, and the discriminant of F/Q. The constant will be conjecturally related to the L-function. Write A for the Néron model of A, and Aw = A ⊗OF kw after a base change to the residue field of F at a finite place w. Write A0w for the connected component of the origin in Aw . Let (ω1 , . . . , ωg ) be an F -basis of H 0 (A, ΩA/F ). Denote the projective OF module of invariant differentials on the Néron model A of A, by ΩA . Then Vg ΩA , restricted to H 0 (A, ΩgA/F ), is given by g ^ ! ωi ·D i=1 for some fractional ideal D of OF . For a complex place w of F , let (h1 , . . . , h2g ) be a Z-basis of the integral 7 homology H1 (A(Fw ), Z). Define Z  Z Ωw := det ωj ωj hi This is non-zero and depends only on . hi Vg i=1 ωi and w. For a real place w of F , corresponding to the embedding σ : F ,→ R, let (h1 , . . . , hg ) be a Z-basis of H1 (A(Fw ), Z)+ , the submodule of the integral homology H1 (A(Fw ), Z) fixed by complex conjugation. Define Aσ (R) Ωw := σ · det A (R)0 This is non-zero and depends only on Z  ωj . hi Vg i=1 ωi and w. Fix bases (x1 , . . . , xn ) and (y1 , . . . , yn ) of respective free subgroups X ⊂ A(F ) and Y ⊂ A(F )∨ of finite index. (Here ∨ indicates the dual abelian variety.) Define the regulator R := | det(< xi , yj >)| A(F ) X ∨ · A Y(F ) , where <, > is the canonical height pairing corresponding to the Poincaré divisor on A × A∨ . The value of R ∈ R∗ is independent of X and Y and their bases. Now we are ready to state the conjecture of Birch and Swinnerton-Dyer: Conjecture 1.0.2 (Birch–Swinnerton-Dyer). We have L(A/F, s) ∼ c(s − 1)rankZ A(F ) as s → 1 8 for some constant c ∈ R∗+ such that c R· Q w|∞ Ωw lies in Q and equals |discF/K |−g/2 · NF/Q D · |III(A)| · Y Aw (kw ) . A0w (kw ) w6 |∞ Note that the last product is well-defined because for all but the finite number of places w of bad reduction, we have Aw (kw ) = A0w (kw ), and the multiplicand equal to 1. We are interested in the case where rankZ A(F ) = 0, whereupon the assertion of the conjecture reduces to Y Aw (kw ) L(A/F, 1) · |A(F )| · |A∨ (F )| Q ∈ Q. = |discF/K |−g/2 ·NF/Q D·|III(A)|· A0w (kw ) w|∞ Ωw w6 |∞ 9 Chapter 2 The Weil restriction. To discuss Gross’ refinement of the Birch–Swinnerton-Dyer Conjecture, we will need the Weil restriction. Let A be abelian variety defined over F , an extension of a number field K. The Weil restriction B := ResFK A is an abelian variety defined over K, whose construction we recall now. For simplicity assume that F/K is Galois. For each element σ of G := Gal(F/K), we have a commutative diagram of schemes Ao  Spec(F ) o MMM MMM MMM M& σ Aσ σ Spec(F )  q qqq q q q qx qq Spec(K) , 10 where Aσ is the fibre product of A  Spec(F ) o σ Spec(F ). Taking the fibre product over all σ, we have B̃ := Q σ∈G A σ  Spec(F )  Spec(K). The finite group G acts on the projective scheme B̃, compatibly with its action on F . Therefore, there exists a scheme B which is the quotient of B̃ by the G-action. B is then defined over K. This variety B is abelian, with the group law inherited from A via B̃. It is seen that B satisfies the desired functorial property: Lemma 2.0.1. B is a K-variety such that HomSpec(K) (Y, B) ≃ HomSpec(F ) (Y ⊗K F, A) for any K-scheme Y . (The functor ResFK is the right adjoint of the extensionof-scalars functor Y 7→ Y ⊗K F . ) In particular, for Y = Spec(C) and a 11 complex place v of K, we have B(C) = HomSpec(K) (Spec(C), B) ≃ HomSpec(F ) (Spec(C⊗K F ), A) = M A(Fw ). w|v If A is an elliptic curve with complex multiplication by an order O in an imaginary quadratic field K, this equality of functors provides B with multiplication by (at least) O. 12 Chapter 3 The conjecture of Gross. Gross has refined (in [9]) the conjecture of Birch–Swinnerton-Dyer for elliptic curves with complex multiplication. Let E be an elliptic curve defined over a number field F , with complex multiplication by the ring of integers OK of a quadratic imaginary field K. If we fix an isomorphism OK ≃ EndF (E), then the action of EndF (E) on Lie(E) gives an embedding K ,→ F . If we fix a complex embedding K ,→ C, we obtain a Grössencharacter ψ of F with values in C∗ . See [21], II, §9. We replace the Q` -action on the `-adic Tate module T` (E) ⊗ Q` by the K ⊗ Q` -action given by complex multiplication. Now the global K-equivariant L-series of E/F is defined K L(E/F, s) := Y K Pw (E/F, Nw −s −1 ) , w6 |∞ where K Pw (E/F, x) = det (1 − xφw | (T` (E) ⊗ Q` )Iw ), K⊗Q` 13 the characteristic polynomial of the action of Frobenius (φw ) upon the inertia `-adic Tate quotient module (T` (A) ⊗ Q` )Iw . The product is over all finite primes of F . On the other hand, we have the Hecke L-series L(ψ, s) := Y (1 − ψ(w)Nw−s )−1 . w6 |∞·fψ (The product is over the finite primes of F not dividing the conductor fψ of ψ.) This product is convergent in the half-plane {s ∈ C : Re(s) > 32 } and has an analytic continuation to the entire plane by Hecke. (See [21], II, §10.) It is known, by a theorem of Deuring, that K L(E/F, s) = L(ψ, s). Hence we also have L(E/F, s) = L(ψ, s)L(ψ, s), reflecting the identity of Euler factors NK/Q (K Pw (x)) = Pw (x). The aim of Gross’ Conjecture is to identify the leading term of L(ψ, s) instead of that of L(E/F, s) (whose leading term is the subject of the Birch–Swinnerton-Dyer Conjecture). 14 3.1 Choice of bases. We continue to write B = ResFK E for the Weil restriction of E. The [F : K]dimensional abelian variety B is canonically self-dual and has complex multiplication by OK . We have K L(B/K, s) = K L(E/F, s). As before, a “constant” is defined using the Néron model, the invariant differentials on E, and their period integrals. But this time, the “constant” is an ideal in OK . This ideal will again be conjecturally related to the L-function. Write B for the Néron model of B, and Bv = B ⊗OK kv after a base change to the residue field of K at a finite place v. Write B0v for the connected component of the origin in Bv . Let (ω1 , . . . , ω[F :K] ) be an K-basis of H 0 (B, ΩB/K ). Denote the projective OK -module of invariant differentials on the Néron model B of B, by ΩB . Then V[F :K] [F :K] ΩB , restricted to H 0 (B, ΩB/K ), is given by  [F :K]  ^ ωi  · d  i=1 for some fractional ideal d of OK . The integral homology H1 (B(C), Z) is a projective OK -module of rank [F : K]. Let (h1 , . . . , h[F :K] ) be a K-basis of H1 (B(C), Q). Then V[F :K] OK H1 (B(C), Z) 15 is given by  [F :K]  ^ hj  · b  j=1 for some fractional ideal b of OK . Let (γ1 , . . . , γ[F :K] ) be the dual K-basis of H 1 (B(C), Q). Then V[F :K] OK H 1 (B(C), Z) is given by   [F :K] ^  γj  · b−1 . j=1 Define Z  ωj . Ω := det hi This is non-zero. Fix a basis (x1 , . . . , xn ) of a free OK -submodule X ⊂ B(K) of finite index. Define the complex regulator RC := det(< xi , xj >C ) B(K) X , where <, >: B(K) × B(K) → R 16 is the height pairing and <, >C : B(K) × B(K) → C is (x, y) 7→ δ < x, y > − < x, δy >, {1, δ} being a Z-basis of OK , and <, >C is independent of the choice of δ. The value of RC ∈ C∗ is independent of X and its basis. 3.2 The conjecture. Let “#” denote the order ideal as an OK -module. Now we can state Gross’ refinement of the Birch–Swinnerton-Dyer Conjecture. Conjecture 3.2.1 (Gross). We have L(ψ, s) ∼ c(s − 1)rankOK B(K) as s → 1 for some constant c ∈ C∗ such that c RC Ω 17 lies in K and generates b · d · #(III(B)) · Y  # w6 |∞ Bv (kv ) B0v (kv )  . We are interested in the case where rankOK B(K) = 0, whereupon the assertion of the conjecture reduces to  L(ψ, s) · |B(K)| Ω  = b · d · #(III(B)) · Y  # w6 |∞ Bv (kv ) B0v (kv )  . As explained by Gross, we have NL(ψ, s) =L(B/K, s), NRC =R, N(Ω) = Y Ωw w|∞ Nb =|discF/K |g/2 Nd =NF/Q D N(#III(B)) =|III(E)|  Y  Bv (kv )  Y Bv (kv ) = . # N B0v (kv ) B0v (kv )  w6 |∞ v6 |∞ (the first three norms being the complex norm z 7→ zz, the others being the norm map on ideals) so that the original Birch–Swinnerton-Dyer Conjecture follows from Gross’ Conjecture. 18 Chapter 4 The language of perfect complexes. In this chapter we reformulate Gross’ Conjecture using the language of perfect complexes and their determinants, essentially following Fontaine and PerrinRiou in [6]. 4.1 Determinants of perfect complexes. By a complex we mean a sequence of modules {M j }j∈Z over some ring, say Q, indexed by the integers, with maps Mj dj / M j+1 satisfying dj+1 ◦ dj = 0 for all j ∈ Z. By a perfect complex we mean a complex which is quasi-isomorphic to a bounded complex whose terms are finite projective modules. (These are the complexes for which it will be possible to define the determinant.) The determinant of a perfect complex is the alternating tensor product 19 of the determinants of the modules M j . Precisely, it is defined as O j (det M j )(−1) , j∈Z where M −1 denotes the dual of M , and det M is the highest nontrivial exterior power of M (if M is nontrivial, and Q if M = 0). In the situations we will be considering, the determinant of the complex is also equal to the alternating tensor product of the determinants of the cohomology modules H j (M • ) of the complex: Lemma 4.1.1. If Q is regular, then det M • = O j (det H j (M • ))(−1) . j∈Z In particular, when Q is a Dedekind ring or a field, the determinant can be computed this way. (See [14].) We will use the following implicitly in calculating the determinants of finitely generated torsion modules. Lemma 4.1.2. Suppose M is a finitely generated torsion module over a discrete valuation ring Q with quotient field F Then the determinant of M is given by: detF (M ⊗Q F ) O O detQ (M ) detF (0) FO O (#M )−1 20 Proof. Write additively the group operation of M . Let x1 , . . . , xn be generators of M , and M 0 the free module on those generators. Let M 00 be the η kernel of M 0 → M , a submodule of M 0 (also free) whose generators we denote by y1 , . . . , yn . Denoting by d : M 00 → M 0 the inclusion, we have d(y1 ) = a11 x1 + · · · + a1n xn , d(y2 ) = a21 x1 + · · · + a2n xn , .. . d(yn ) = an1 x1 + · · · + ann xn , for some ajk ∈ Q, j, k = 1, . . . , n. Then M has the finite projective resolution ··· /0 / M 00 / M0 /M /0 / ··· , concentrated in degrees -1 and 0. So det M = det M 0 ⊗ det−1 M 00 . Over F , d induces the isomorphism M 00 ⊗Q F ≃ d / M 0 ⊗Q F . We wish to identify the invertible Q-module det M 0 ⊗Q (det M 00 )−1 21 inside the invertible F -module   −1  det(M 0 ⊗Q F ) ⊗F det(M 00 ⊗Q F ) . F F Now    −1 00 det(M ⊗Q F ) ⊗F det(M ⊗Q F ) 0 F F has canonical basis d(y1 ) ∧ · · · ∧ d(yn ) ⊗ (y1 ∧ · · · ∧ yn )−1 while det M 0 ⊗Q (det M 00 )−1 has canonical basis det(ajk )−1 · d(y1 ) ∧ · · · ∧ d(yn ) ⊗ (y1 ∧ · · · ∧ yn )−1 . We have the diagram η(y1 ) ∧ · · · ∧ η(yn ) ⊗ (y1 ∧ · · · ∧ yn )−1  /1 ∈ ∈ (detF (M 0 ⊗Q F )) ⊗F (detF (M 00 ⊗Q F ))−1 O O O / Q · det(ajk )−1 ∈ ≃ /F O ∈ det M 0 ⊗Q (det M 00 )−1 ≃ det(ajk )−1 · η(y1 ) ∧ · · · ∧ η(yn ) ⊗ (y1 ∧ · · · ∧ yn )−1  / det(ajk )−1 , 22 where the isomorphisms are induced by d. As Q is a DVR, det(ajk ) · Q = #M . The result follows. 4.2 The setting. Let B be an abelian variety defined over the quadratic imaginary field K, and so that there is an embedding OK ,→ EndK (B). Denote by B ∨ the dual abelian variety. The situation we have in mind is that in which B is the Weil restriction of an elliptic curve with complex muliplication by OK . Here we would have B ≃ B ∨ . We make the same assumption about the Tate–Šafarevič group as before: Assumption 4.2.1. III(B) is finite. Fix K ,→ C. 4.3 Four vector spaces. We have four finite-dimensional K-vector spaces H 0 (B, Ω1 ), H1 (B(C), Q), B(K) ⊗Z Q, B ∨ (K) ⊗Z Q and perfect K ⊗ R = C-linear pairings (H 0 (B, Ω1B/K ) ⊗Q R) × (H1 (B(C), Q) ⊗Q R) → C (4.1) induced by integration and (B(K) ⊗Z R) × (B ∨ (K)c ⊗Z R) → C (4.2) 23 given by the modified height pairing <, >C discussed earlier. (For any Kmodule W we denote by W c the same group W with the conjugate action of K.) Defining K Ξ := det H K 0 (B, Ω1B/K ) ⊗K det H1 (B(C), Q) K ⊗K (det B(K) ⊗Z Q) ⊗K (det B ∨ (K)c ⊗Z Q), K K these pairings induce a K ⊗ R = C-linear isomorphism K ϑ∞ : K Ξ ⊗Q R ≃ K ⊗ R. 4.4 Galois cohomology. We fix a prime number p, and a finite set S of places of K, containing the infinite place, those places above p, and all places where B has bad reduction. We denote by GS the Galois group of the maximal extension of K unramified outside S and by OK,S the ring of S-integers of K. For any continuous GS module N we put RΓ(OK,S , N ) = C • (GS , N ), the standard complex of continuous cochains. We define RΓc (OK,S , N ) ! =Cone RΓ(OK,S , N ) → M v∈S RΓ(Kv , N ) [−1]. 24 Suppose v 6 | p is a place of K. Define RΓf (Kv , Vp (B)) to be the perfect complex, concentrated in degrees 0 and 1, of K ⊗ Qp -modules /0 ··· / V (B)Iv1−Frobv/ V (B)Iv p p /0 / ··· . (4.3) Define RΓf (Kv , Tp (B)) to be the complex \ B(K v )[−1], which is ··· /0 / \ /0 B(Kv ) / ··· , (4.4) n \ concentrated in degree 1, where B(K v ) := limB(Kv )/p is the p-completion of ←− n B(Kv ). For v | p, we define RΓf (Kv , Vp (B)) to be the perfect complex, concentrated in degrees 0 and 1, of K ⊗ Qp -modules ··· /0 /D v (1−φ,π) / Dv ⊕ H 0 (B, Ω1 ∨ B/Kv ) /0 / ··· , (4.5) 25 where Dv denotes Dcris (Vp (B)) = H 0 (Kv , Bcris ⊗Qp Vp (B)) and ∨ denotes the K ⊗ Qp -dual. The Kv -vector space H 0 (Kv , Bcris ⊗Qp Vp (B)), though invariant under the action of Gal(Kv /Kv ), has an additional Frobenius automorphism, which is what we mean here by φ. Define RΓf (Kv , Tp (B)) to be \ B(K v )[−1]. First, as in [2], we have a natural map of complexes of OK,p -modules RΓf (Kv , Tp (B)) → RΓ(Kv , Tp (B)) and denote by RΓ/f (Kv , Tp (B)) the cone of this map. Whether v 6 | p or v | p, we have compatibility between RΓ(Kv , Tp (B)), RΓ(Kv , Vp (B)), 26 and the natural map RΓf (Kv , Tp (B)) → RΓ(Kv , Tp (B)), as follows. If v 6 | p, then 0 (K )[−1] ⊗ RΓ(Kv , Tp (B)) ⊗Zp Qp = B\ v Zp Qp is the zero complex, which is quasi-isomorphic to RΓ(Kv , Vp (B)). If v | p, we have \ RΓf (Kv , Tp (B)) ⊗Zp Qp = B(K v ) ⊗Zp Qp [−1], which is quasi-isomorphic to the complex RΓf (Kv , Vp (B)), via the exponential map H 0 (B, ΩBv )∨ in degree 1. ≃ / \ exp B(Kv ) ⊗Zp Qp 27 Remark 4.4.1. If B has good reduction at v 6 | p, then RΓf (Kv , Tp (B)) could have been defined as was RΓf (Kv , Vp (B)) to obtain the same cohomology. Precisely, 0 \ \ RΓf (Kv , Tp (B)) = B(K v ) = B (Kv ) is quasi-isomorphic to the complex ··· /0 / T (B)Iv1−Frobv/ T (B)Iv p p /0 concentrated in degrees 0 and 1, which lies inside RΓf (Kv , Vp (B)). The quasi-isomorphism is compatible with the natural map RΓf (Kv , Tp (B)) → RΓ(Kv , Tp (B)), / ··· , (4.6) 28 whose mapping cone we denote RΓ/f (Kv , Tp (B)). Second, we define RΓf (K, Tp (B))   M =Cone RΓ(OK,S , Tp (B)) → RΓ/f (Kv , Tp (B)) [−1]. v∈S−{∞} Third, we recall that the complex RΓc (OK,S , Tp (B)) by definition fits into the exact triangle RΓc (OK,S , Tp (B)) → RΓ(OK,S , Tp (B)) → M RΓ(Kv , Tp (B)). v∈S Using the complexes defining RΓ/f (Kv , Tp (B)), RΓf (K, Tp (B)), and RΓc (OK,S , Tp (B)), we obtain a distinguished triangle  RΓc (OK,S , Tp (B)) → RΓf (K, Tp (B)) →   M RΓf (Kv , Tp (B)) ⊕ RΓ(C, Tp (B)). v∈S−{∞} (4.7) We define RΓ/f (Kv , Vp (B)), 29 RΓf (K, Vp (B)), RΓc (OK,S , Vp (B)), similarly. There is a version of triangle (4.7) with rational (Vp (B)) coefficients, and we have RΓc (OK,S , Tp (B)) ⊗Zp Qp ≃ RΓc (OK,S , Vp (B)). Here, since we are dealing with the regular rings K, K ⊗ Qp , and OK ⊗ Zp , the determinants of the complexes considered are alternating tensor products of the determinants of the cohomology modules, as noted in Chapter 4.1. 4.5 Reformulation of Gross’ Conjecture. In order to reformulate Gross’ Conjecture, we wish to relate #(III(B)p∞ ) to the determinant of RΓc (OK,S , Tp (B)), viewed as an integral lattice inside the determinant of RΓc (OK,S , Vp (B)). 30 From the distinguished triangle (4.7), it will suffice to compute the determinants of v ∈ S − {∞} RΓf (Kv , Tp (B)), (4.8) RΓf (K, Tp (B)), (4.9) RΓ(C, Tp (B)), (4.10) describing them as lattices in the respective invertible Kp -modules det RΓf (Kv , Vp (B)), Kp det RΓf (K, Vp (B)), Kp det RΓ(C, Vp (B)). Kp The determinant of the first complex (4.8) is given by the following lemma. Lemma 4.5.1. For RΓf (Kv , Tp (B)) ⊂ RΓf (Kv , Vp (B)), v finite, we have  det RΓf (Kv , Tp (B)) =# OK,p Bv (kv ) B0v (kv )  ⊂Kp ≃ det RΓf (Kv , Vp (B)) Kp (4.11) (4.12) 31 if v 6 | p, and (4.13)    [F :K] ^ Bv (kv )  det RΓf (Kv , Tp (B)) =d · # · ωj  OK,p B0v (kv ) j=1   [F :K] ^ ⊂Kp ·  ωj  ≃ det RΓf (Kv , Vp (B))  Kp j=1 (4.14) (4.15) if v | p. Here the isomorphisms are induced by maps between cohomology in degrees 0 and 1. For the proof of the lemma we use the following two results, The first is from Lemma 1 of [3], and the second is proved using [11]. Lemma 4.5.2. Suppose ··· /M /0 η /M /0 / ··· , is a complex of finite projective Q-modules, concentrated in degrees 0 and 1. Then there is a commutative diagram det(Q) det(M ) ⊗  det(M )−1 ≃ /Q Lemma 4.1.1  detQ H (M ) ⊗ detQ H 1 (M )−1 0 detQ (0) ⊗ detQ (0)−1 where the horizontal isomorphism is induced by idM . det(η)  Q 32 Lemma 4.5.3. 1 Iv 0 \ B(K v )/H (kv , Tp (B) ) ≃ Bv (kv )/B (kv ). Proof. We denote by B the Néron model of B/Kv , by B0 the zero connected component subscheme, by Bv the special fibre, and by B0v the zero connected component of Bv . By [11], Lemma 2.2.1, there is a short exact sequence of group schemes / B0 / B0 n 0 p pn / B0 /0. We also use the short exact sequence of group schemes / Bpn 0 /B pn /B /0. From these exact sequences for B0 and B, we obtain the exact sequences B(Kv )pn pn / B(Kv )pn B0 (OK,v )pn pn / B0 (OK,v )pn B0v (kv )pn pn / B0 (kv )pn v / H 1 (Kv , Bpn ) , / H 1 (OK,v , B0n ) , p / H 1 (kv , (B0 )pn ) , v which induce respective injections B(OK,v )pn B(Kv )pn / / H 1 (Kv , Bpn ) H 1 (Kv , Bpn ) , 33 B0 (OK,v )pn / B0 (kv )pn / / H 1 (OK,v , B0n ) , p / H 1 (kv , i∗ B0n ) H 1 (kv , i∗ (B0v )pn ) , p where i here denotes the natural map Speckv / / SpecOK,v . Taking projec- tive limits, we obtain, respectively, / \ B(K v) \ B(O K,v ) / 0 (O B\ K,v ) 0 (k ) / \ B v / H 1 (Kv , Tp B) H 1 (Kv , Tp (B)) , / H 1 (OK,v , Tp (B0 )) , / H 1 (kv , i∗ Tp (B0 )) H 1 (kv , i∗ Tp (B0v )) . Together, these form a diagram: \ B(O K,v ) \ B(K )/ O v O / H 1 (Kv , Tp B) O / 0 (O B\ K,v )  0 (k ) / \ B v H 1 (Kv , Tp (B)) / H 1 (OK,v , Tp (B0 )) base change / H 1 (kv , i∗ Tp (B0 )) H 1 (kv , i∗ Tp (B0v )) The injection B0 (kv )pn / / H 1 (kv , i∗ B0n ) p H 1 (kv , i∗ (B0v )pn ) , fits into the exact sequence B0 (kv )pn / / H 1 (kv , i∗ (B0 )pn ) v / H 1 (k , B0 ) . v v 34 By Lang’s Theorem H 1 (kv , B0v ) = 0, so that the map B0 (kv )pn / / H 1 (kv , i∗ B0n ) H 1 (kv , i∗ (B0v )pn ) . p is surjective. By [11], Proposition 2.2.5, we have H 1 (kv , i∗ Tp (B0v )) = H 1 (kv , (Tp (B 0 ))Iv ), The map 0 (O 0 \ B\ K,v ) → B (kv ) is surjective since A0 → OK,v is smooth and OK,v Henselian, injective since v 6 | p, making ker(B0 (OK,v ) → B0 (kv )) p-divisible. Since Speckv is proper over SpecOK,v , then H 1 (OK,v , Tp (B0 )) ≃ H 1 (kv , i∗ Tp (B0 )). We obtain this diagram: \ B(O K,v ) / \ B(K v) O O / 0 (O B\ K,v )   0 (k ) / \ B v / H 1 (Kv , Tp B) O H 1 (Kv , Tp (B)) / H 1 (OK,v , Tp (B0 )) ≃ / / H 1 (kv , i∗ Tp (B0 )) H 1 (kv , i∗ Tp (B0v )) H 1 (kv , (Tp (B))Iv ) 35 From this, we conclude that 1 Iv 0 \ \ \ \0 B(K v )/H (kv , Tp (B) ) ≃ B(OK,v )/B (OK,v ) ≃ Bv (kv )/B (kv ). Since Bv (kv )/B0 (kv ) is finite, this last equality becomes 1 Iv 0 \ B(K v )/H (kv , Tp (B) ) ≃ Bv (kv )/B (kv ). Proof of Lemma 4.5.1 First suppose v 6 | p. By Lemma 4.5.2 and the definition of RΓf (Kv , Tp (B)), we see that   −1 \ det RΓf (Kv , Tp (B)) =# B(K v ) · det(1 − Frobv ) OK,p To calculate det(1 − Frobv )−1 , we use the exact sequence Tp (B)Frobv =1 / 0 / Tp (B)1−Frobv/ Tp (B) / / H 1 (k , T (B)Iv ), v p 36 which shows that det(1 − Frobv )−1 = #H 1 (kv , Tp (B)Iv ). We must now compute   1 Iv \ # B(K )/H (k , T (B) ) . v v p But this is given by Lemma 4.5.3 to be  # Bv (kv ) B0v (kv )  . Now suppose v | p. The term H 0 (B, Ω1B/Kv )∨ in degree 1 contributes a factor of  [F :K]  det H 0 (B, Ω1B/Kv ) = d ·  ^ ωj  OK,p j=1 to det RΓf (Kv , Tp (B)), OK,p and the factor  # Bv (kv ) B0v (kv )  37 is obtained as in the case v 6 | p. To deal with the second complex (4.9), we use the following lemma, obtained from the computations of Burns–Flach in [2]. Lemma 4.5.4. The cohomology of RΓf (K, Tp (B)) is given by Hf0 (K, Tp (B)) =0, (4.16) Hf1 (K, Tp (B)) =B(K) ⊗Z Zp , (4.17) 0 → III(B)p∞ →Hf2 (K, Tp (B)) → HomZp (B ∨ (K)c , Zp ) → 0 (exact), (4.18) Hf3 (K, Tp (B)) =(B ∨ (K)p∞ )∧ , (4.19) where ∧ denotes the Pontryagin dual. Proof. From the definition of RΓf (K, Tp (B)), it is apparent that Hf0 (K, Tp (B)) = H 0 (OK,S , Tp (B)), which is 0. This is the statement (4.16). It is also apparent that  Hf1 (K, Tp (B)) = ker H 1 (OK,S , Tp (B)) → M v∈S−{∞} 1 H (Kv , Tp (B)) ⊕ H 1 (C, Tp (B)) . Hf1 (Kv , Tp (B)) Since C is algebraically closed, H 1 (C, Tp (B)) = 0. Since for all v, \ Hf1 (Kv , Tp (B)) = B(K v ), then the above kernel must contain B(K) ⊗Z Zp .  38 Since III(B), which is finite, surjects onto the kernel of H 1 (OK,S , Vp (B)/Tp (B)) → Hf1 (K, Tp (B)) ⊗ Qp /Zp M v∈S−{∞} H 1 (Kv , Vp (B)/Tp (B)) , Hf1 (Kv , Tp (B)) ⊗ Qp /Zp B(K) ⊗Z Zp is in fact all of Hf1 (K, Tp (B)). This is statement (4.17). To prove the statement (4.18), we use four pairings. The Weil pairing is Tp (B) × Tp (B ∨ ) / Zp (1) . Second, we have the Pontryagin pairing / Qp /Zp (1) . Tp (B) × Bp∨∞ Vp (B ∨ )/Tp (B ∨ ) Third, we have Poitou-Tate duality / H 3 (OK,S , Qp /Zp (1)) Hc2 (OK,S , Tp (B)) × H 1 (OK,Sp , Vp (B ∨ )/Tp (B ∨ )(1)) c trace  Qp /Zp . We obtain 0 / H 2 (K, T (B))∧ f p / H 1 (OK,S , B ∨∞ ) p / L H 1 (Kv ,Bp∨∞ ) v∈S−{∞} Hf1 (Kv ,Tp (B))⊥ . Then we see that Hf2 (K, Tp (B))∧ is just the Selmer group Sel(B ∨ )p∞ , and Hf1 (Kv , Tp (B))⊥ = (B(Kv ))⊥ , which is B ∨ (Kv ) ⊗ Qp /Zp . Consider the Galois 39 cohomology exact sequence 0 → B ∨ (K) ⊗ Qp /Zp → Sel(Bp∨∞ ) → III(Bp∨∞ ) → 0. (4.20) We use a fourth pairing, the Cassels–Tate pairing, which says that III(Bp∨∞ )∧ ∼ III(B)p∞ . Also, we know that the Pontryagin dual of B ∨ (K)⊗Qp /Zp is HomZp (B ∨ (K), Zp ). Therefore, dualizing (4.20), we find that 0 → III(B)p∞ → Hf2 (K, Tp (B)) → HomZp (B ∨ (K), Zp ) → 0. The statement (4.19) follows from Tate–Poitou duality: Hf3 (K, Tp (B)) ≃Hc3 (OK,S , Tp (B)) ≃H 0 (OK,S , Vp∨ (B)/Tp∨ (B)(1))∧ ≃(B ∨ (K)p∞ )∧ From this lemma we deduce a corollary which identifies the rational cohomology. 40 Corollary 4.5.5. The cohomology of RΓf (K, Vp (B)) is given by Hf0 (K, Vp (B)) =0, (4.21) Hf1 (K, Vp (B)) ≃B(K) ⊗Z Qp , (4.22) Hf2 (K, Vp (B)) ≃(B ∨ (K)c ⊗Z Qp )∨ , (4.23) Hf3 (K, Vp (B)) =0, (4.24) where c denotes the conjugate Gal(K/K)-action. The following lemma describes (4.10), the last of the three complexes. Lemma 4.5.6. For RΓ(C, Tp (B)) ⊂ RΓ(C, Vp (B)), we have:  [F :K]  det H 0 (C, Tp (B)) =b−1 ·  ^ γj  OK,p (4.25) j=1  [F :K] ⊂Kp ·  ^ j=1  γj  = det H 0 (C, Vp (B)), Kp det H j (C, Tp (B)) =OK,p (4.26) (4.27) OK,p ⊂Kp = det H j (C, Vp (B)) for j 6= 0. Kp (4.28) Proof. This result follows from the definition of b. The last thing we need to translate Gross’ Conjecture is a map relating K Ξ to the determinant of the complex RΓc (OK,S , Vp (B)). Lemma 4.5.7. There is a K ⊗Q Qp -linear isomorphism induced by the trian- 41 gle (4.7): −1 K ϑp : detK⊗Qp RΓc (OK,S , Vp (B)) ≃ / K Ξ ⊗Q Qp . Proof. The result will follow immediately from the triangle (4.7) if we can show that there is a natural K ⊗Q Qp -linear isomorphism det RΓf (K, Vp (B))−1   M ⊗ det  RΓf (Kv , Vp (B)) K⊗Qp K⊗Qp v∈S−{∞} ⊗ det RΓ(C, Vp (B)) K⊗Qp ≃ →K Ξ ⊗Q Qp . Recall that K Ξ = det H K 0 (B, Ω1B/K ) ⊗K det H1 (B(C), Q) K ⊗K (det B(K) ⊗Z Q) K ⊗K (det B ∨ (K)c ⊗Z Q). K 42 First, H 0 (C, Vp (B)) ≃H1 (B(C), Q) ⊗ Qp , H j (C, Vp (B)) =0 for j 6= 0, whence det RΓ(C, Vp (B)) = det (H1 (B(C), Q) ⊗ Qp ). K⊗Qp K⊗Qp Second, for v | p, Hf1 (Kv , Vp (B))∨ exp∗ / H 0 (B, Ω1B/K ), ≃ Hfj (Kv , Vp (B) = 0 for j 6= 1, the isomorphism being given by dual of the exponential map discussed in Section 4.4, whence  det  K⊗Qp  M RΓf (Kv , Vp (B)) ≃ det H 0 (B, Ω1B/K ), K v|p while for v 6 | p, RΓf (Kv , Vp (B)) is acyclic. 43 Third, Lemma 4.5.5 says that Hf1 (K, Vp (B)) =B(K) ⊗Z Qp , Hf2 (K, Vp (B)) =(B ∨ (K)c ⊗Z Qp )∨ , Hfj (K, Vp (B)) =0 for j 6= 1, 2, whence det RΓf (K, Vp (B))−1 =( det B(K) ⊗Z Qp ) K⊗Qp K⊗Qp ⊗K⊗Qp ( det B ∨ (K)c ⊗Z Qp ). K⊗Qp We obtain the K ⊗Q Qp -linear isomophism det RΓf (K, Vp (B))−1   M ⊗ det  RΓf (Kv , Vp (B)) K⊗Qp K⊗Qp v∈S−{∞} ⊗ det RΓ(C, Vp (B)) K⊗Qp ≃ →K Ξ ⊗Q Qp and the result follows. Now we can reformulate the p-part of Gross’ Conjecture. Theorem 4.5.8. The p-part of Gross’ Conjecture is equivalent to the statement that L(ψ, s) ∼ c(s − 1)rankOK B(K) as s → 1 44 for some constant c ∈ C∗ such that −1 K ϑ∞ (c) ∈ K Ξ ⊗ 1 and that −1 −1 OK,p · K ϑ−1 p (K ϑ∞ (c)) = det RΓc (OK,S , Tp (B)) . OK,p Proof. Let us examine the isomophism K ϑ∞ . From the pairings (4.1) and (4.2), we find that the image under K ϑ∞ of the generator of K Ξ given by the chosen bases ({γj }j , {ωj }j , and {xj }j ), is Ω · det (< xj , xk >)j,k = ΩRC OK,p B(K) . X So the statements −1 K ϑ∞ (c) ⊂ K Ξ ⊗ 1 and −1 −1 OK,p · K ϑ−1 p (K ϑ∞ (c)) = det RΓc (OK,S , Tp (B)) OK,p are equivalent to c RC Ω ∈K (4.29) and OK,p · K ϑ−1 ∞  c RC Ω   = K ϑp −1 det RΓc (OK,S , Tp (B)) OK,p  (4.30) 45 (using for the second equality the fact that B ∨ (K) ≃ B(K)). The first statement of Gross’ Conjecture is just inclusion (4.29), and the p-part of the second is exactly  c   RC Ω = b · d · #(III(B)) · p Y  # v6 |∞  Bv (kv )  , B0v (kv )  (4.31) p or, equivalently, OK,p · K ϑ−1 ∞  c RC Ω   Y b · d · #(III(B)) · = OK,p · K ϑ−1 ∞ v6 |∞  #  Bv (kv )  . B0v (kv )  (4.32) Comparing (4.30) and(4.32), we see from the triangle (4.7) what must be proven: that  K ϑp −1  det RΓc (OK,S , Tp (B)) OK,p is generated, in detRΓf (K, Vp (B))−1 Kp   M ⊗ det  RΓf (Kv , Vp (B) Kp v∈S−{∞} ⊗ detRΓ(C, Vp (B)), Kp by b · d · #(III(B)) · Y v6 |∞  # Bv (kv ) B0v (kv )  , 46 with respect to the chosen bases. But this follows from Lemma 4.5.1, Lemma 4.5.4 and its Corollary 4.5.5, and Lemma 4.5.6. 47 Chapter 5 Elliptic curves with nonmaximal endomorphism ring. In order to study an elliptic curve E with nonmaximal endomorphism ring, and its Weil restriction B, we employ another elliptic curve isogenous to E and defined over the same base field F as E. Lemma 5.0.1 ([21], Exercise II.1.2). Suppose E is an elliptic curve over a (number) field F with EndF (E) ≃ O ⊆ OK . Then there exists an elliptic curve E0 over F with EndF (E0 ) ≃ OK , and an isogeny E → E0 over F . Proof. Say that OK is generated over Z by 1 and α, and that O = Z + f · OK = Z + f α · Z. Define E0 by the following commutative diagram, with exact row and di- 48 agonal: / Ef 0 / E [f ] / E @@ @@ @@ @@ [f α]  EA AA AA AA E0 @ @@ @@ @@ @ 0 We shall construct an OK -action on E0 which agrees with the O-action on E . We construct morphisms υ1 , υ2 , υ3 , as shown in the following diagram: Ef 0 / Ef AA AA AA AA /E /E AA AA AA AA  EB υ1 AA AA AA AA E0 ? υ2 BB BB BB B  ~ ?? ?? ?? ? υ3 E0 B BB BB BB B! 0 0 The upper diagonal is the same as the lower diagonal. The morphism υ1 is defined by commutativity. The morphism υ2 is obtained by observing that the kernel of the morphism E [f ] /E is contained in the kernel of υ1 . The morphism υ3 is obtained as follows. The 49 kernel of the upper diagonal morphism EA AA AA AA E0 is the image of the composite morphism: /E Ef  [f α] E This image is [f α]Ef , which has pre-image [f α]Ef 2 under [f ] E /E, whose image under E  [f α] E 50 is [f α]2 Ef 2 =[f 2 α2 ]Ef 2 =[f α2 ]Ef =[f (aα + b)]Ef 2 for some a, b ∈ Z =([a][f α] + [b][f ])Ef =[a][f α]Ef , which is contained in the kernel of the lower diagonal morphism: EA AA AA AA E0 Therefore, the kernel of the upper diagonal morphism EA AA AA AA E0 (in which we were originally interested) is contained in the kernel of υ2 , and υ3 can be constructed making the diagram commute. We then define [α] : E0 → E0 51 to be υ3 . The morphisms 1 = id : E0 → E0 and [α] : E0 → E0 are extended by linearity to an action of OK upon E0 . One checks that this action is a well-defined ring action and agrees with the action of O upon E. That is, for any x ∈ O ⊂ OK , the diagram E  E0 [x] [x] /E  /E 0 commutes. It is known (see for example [20], Appendix C, Theorem 11.5) that the j-invariant of E (resp. E0 ) belongs to the ring class field H(O) associated to O (resp .the Hilbert class field H = H(OK ) of K). We have a tower of fields K ⊂ H ⊂ H(O) ⊂ F . The curves E and E0 are twists over F of elliptic curves defined over H(O) and H, respectively. As before, we have B the Weil restriction of E, and we define B0 to be the Weil restriction of E0 . The varieties B and B0 are isogenous abelian varieties over K with complex multiplication by O and OK , respectively. 52 5.1 The Burns–Flach Conjecture. The elliptic curve E is defined over F , with complex multiplication by the order O ⊆ OK , with Weil restriction B := ResFK E, and with E having Grössencharacter ψ. The Burns–Flach Conjecture concerns the Weil restriction B of the original elliptic curve E. Let f be the conductor of F . Fix a prime p. Let S, a set of places of K, be defined as the union of the set of prime divisors of fp, and the set of primes below those of bad reduction of E/F , along with the infinite place. Then by the Criterion of Néron–Ogg–Šafarevič, S − {∞} is the set of places where the extensions F (Epn )/K are ramified. Conjecture 5.1.1 (Burns–Flach). We have L(ψ, s) ∼ c(s − 1)rankOK B(K) as s → 1 for some constant c ∈ C∗ such that −1 K ϑ∞ (c) ∈ K Ξ ⊗ 1 and −1 −1 Op · K ϑ−1 p (K ϑ∞ (c)) = det RΓc (OK,S , Tp (B)) . Op (5.1) Note 5.1.2. The Burns–Flach Conjecture is a strengthening of the p-part of Gross’ Conjecture (for primes p dividing the conductor of O). Taking tensor 53 products with OK,p in the statement of the Burns–Flach Conjecture, we have Tp (B) ⊗Op OK,p = Tp (B0 ), and the determinant over Op becomes a determinant over OK,p , yielding the p-part of the statement of Gross’ Conjecture. 5.2 A refinement in the case F (Etors)/K is abelian. We wish to refine the conjecture further by considering the full endomorphism algebra of B, which can be much larger than K. Write R := EndK (B), R := R ⊗Z Q, Rp := R ⊗Z Zp , and Rp := R ⊗Z Qp = R ⊗Q Qp . Lemma 5.2.1. Suppose F (Etors )/K is abelian. Then R is a commutative semisimple K-algebra of rank [F : K]. Also, R is projective over O and Tp (B) is projective over Rp . Proof. R is computed in [8] Section 4. R = EndK (B) is the Galois invariants of EndF (B), where B is isomorphic over F to Y Eσ. σ∈G Therefore R is the Galois invariants of ! HomF Y σ∈G Eσ, Y σ∈G Eσ = Y Hom(E σ1 , E σ2 ), σ1 ,σ2 ∈G which is seen to be Y Hom(E σ , E) · σ. σ∈G Thus R is projective over O. As well, as in [8], §4, R is a commutative semisim- 54 ple K-algebra. (In the special case where E is isogenous to a base change from K to F (as in the examples we construct later), R ≃ K[G].) We have Tp (B) = Y Tp (E τ ), τ ∈G induced by the decomposition B(K) = E(K ⊗K F ) = E( Y Y K) = τ ∈G E τ (K). τ ∈G We know that Tp (E) is a free Op -module of rank 1. Pick an Op -basis ξ of Tp (E). And for each σ ∈ G, HomOp (E σ , E) ⊗Z Zp is also a free Op -module of rank 1. Pick an Op -basis b(σ) for HomOp (E σ , E) ⊗Z Zp , that is, an isogeny b(σ) E σ −→ E of degree prime to p. Composing σ with b(σ), we obtain (since b(σ) has a kernel prime to p) a map E(K ⊗K F ) / E σ (K ⊗ σ K F) b(σ) / E(K ⊗ K F) . Or, in more detail, where B σ = ResFK E σ , we obtain Q τ τ ∈G E (K) / B(K) = / E σ (K ⊗ σ K F) b(σ) / E(K ⊗ K F) σ / Q τσ τ ∈G E (K) ≃ ≃ ≃ E(K ⊗K F ) b(σ) = / B σ (K) σ = B(K) b(σ)τ / Q τ ∈G E τ (K). The isogeny b(σ) induces an isomorphism Tp (B σ ) ≃ Tp (B). So, taking Tate 55 modules, we obtain the diagram / Tp (B) Q τ ∈G Tp (E τσ b(σ) ) / ( ξ , 0, . . . , 0)  / (0, . . . , 0, ξ , 0, . . . , 0)  τ ∈G Tp (E τ ) / (0, . . . , 0, b(σ)σ−1 ξ, 0, . . . , 0) τ =σ −1 τ =1 Q ∈ / σ ) ∈ τ = = = τ ∈G Tp (E ∈ Q b(σ) / Tp (B σ ) σ Tp (B) τ =σ −1 −1 Since ξ is an Op -basis of Tp (E), then b(σ)σ ξ is also an Op -basis of Tp (E). Suppose −1 (ατ · b(τ )τ ξ)τ ∈G is any element of Y Tp (E τ ) = Tp (B). τ ∈G We can write ! −1 (ατ · b(τ )τ ξ)τ ∈G = X ατ · b(τ ) · τ ( ξ , 0, . . . , 0) τ =1 τ ∈G with unique ατ ∈ Op , that is, with unique ! X ατ · b(τ ) · τ ∈ Rp . τ ∈G Therefore ( ξ , 0, . . . , 0) is an Rp -basis of Tp (B). In particular, Tp (B) is a τ =1 projective Rp -module. 56 We define an L-function for B as we did for E, R L(B/K, s) := Y R Pw (B/K, Nw −s −1 ) : C → R ⊗Q C, w6 |∞ where R Pw (B/K, x) = det (1 − xφw | (T` (B) ⊗ Q` )Iw ). R⊗Q` Write ϕ for a Grössencharacter of K, which when pre-composed with the b norm, gives ψ. Then, as in [8], the possible choices for ϕ are {ϕχ : χ ∈ G}. Writing X for the set of orbits of {ϕχ : χ ∈ Ĝ} under AutK C, we have R = K ⊗O R = K ⊗O EndK B ≃ Y K(ϕχ), X where K(ϕχ) is K with the values of ϕχ adjoined. The L-function R L(B, s) corresponds to the tuple of functions (L(σ(ϕχ), s))X,σ∈HomK (K(ϕχ),C) , which is considered an element of R ⊗Q C, where R≃ Y X K(ϕχ). 57 Define R Ξ := det H R 0 (B, Ω1B/K ) ⊗R det H1 (B(C), Q) R ⊗R (det B(K) ⊗Z Q) ⊗R (det B ∨ (K)c ⊗Z Q), R R As with K ϑ∞ in Section 4.3, the integration and modified height pairings induce a R ⊗ R-linear isomorphism R ϑ∞ : R Ξ ⊗Q R ≃ R ⊗ R. We have a R ϑp , which is the analog of K ϑp : Lemma 5.2.2. There is a R ⊗Q Qp -linear isomophism: R ϑp : ≃ det RΓc (OK,S , Vp (B))−1 → R Ξ ⊗Q Qp . R⊗Qp Proof. This is proved in exactly the same way as Lemma 4.5.7. Now we can state the refined Burns–Flach Conjecture: Conjecture 5.2.3. We have rankR B(K) as s → 1 R L(B, s) ∼ R c(s − 1) for some R c ∈ (R ⊗ R)∗ such that −1 R ϑ∞ (R c) ⊂ R Ξ ⊗ 1 58 and −1 −1 Rp · R ϑ−1 p (R ϑ∞ (R c)) = det RΓc (OK,S , Tp (B)) . Rp (5.2) Note 5.2.4. Taking norms from Rp to Op (which is possible by Lemma 5.2.1) yields the original conjecture of Burns–Flach. 59 Chapter 6 Translation into the terminology of Kato. In this chapter we pose a conjecture in the terminology of Kato, which is equivalent to the refined Burns–Flach Conjecture for B. Assumption 6.0.1. We have rank(E(F )) = 0, or equivalently rank(B(K)) = 0. In particular, this makes R Ξ = det H R 0 (B, Ω1B/K ) ⊗R det H1 (B(C), Q). R 60 For ease of notation, we will now write the following: T :=Tp (B) V :=Vp (B) Ξ :=R Ξ ϑ∞ :=R ϑ∞ ϑp :=R ϑp c :=R c The Rp -sheaf T on SpecOK,S is smooth and invertible. Define the invertible Rp -module ∆Rp (OK,S , T ) to be  −1  −1 det(RΓ(OK,S , T )) ⊗Rp det(RΓ(OK,S ⊗Z R, T (−1))) , Rp Rp and ∆Rp (OK,S , V ) to be ∆Rp (OK,S , T ) ⊗Zp Qp We compute the cohomology of RΓ(OK,S , V ) as follows: Lemma 6.0.2. • H 1 (OK,S , V ) is a 1-dimensional Rp -vector space. 61 • H j (OK,S , V ) = 0 for j 6= 1. Proof. Use the exact triangle Rf Γ(OK,S , V ) → RΓ(OK,S , V ) → M R/f Γ(Kv , V ) v∈S (see [4], §3.2), which is the complex 0 →Hf0 (K, V ) → H 0 (OK,S , V ) → M H/f0 (Kv , V ) v∈S →Hf1 (K, V ) → H 1 (OK,S , V ) → M →Hf2 (K, V ) → H 2 (OK,S , V ) → M H/f1 (Kv , V ) v∈S H/f2 (Kv , V ) v∈S →Hf3 (K, V ) → · · · . Now H 0 (OK,S , V ) = 0, and since RΓf (K, V ) is acyclic by Assumption 6.0.1, we have H 1 (OK,S , V ) ≃ M H/f1 (Kv , V ), v∈S H 2 (OK,S , V ) ≃ M v∈S H/f2 (Kv , V ), 62 the second giving us by duality H 2 (OK,S , V ) ≃ M ≃ M ≃ M H/f2 (Kv , V ) v∈S H 2 (Kv , V ) v∈S H 0 (Kv , V ∗ (1))∗ v∈S ≃ M H 0 (Kv , V )∗ v∈S since V is the Weil restriction of an elliptic curve =0. Consider the exact sequence 0 / H 1 (Kv , V ) f / H 1 (K , V ) v / H 1 (Kv , V ) /f 0 If v 6 | p, then H 1 (Kv , V ) = 0, and for v | p, we compute dimRp H 1 (OK,S , V ) = X = X dimRp H/f1 (Kv , V ) v|p v|p dimRp H 1 (Kv , V ). /0. 63 Using Kp = Q v|p Kv we rewrite this: dimOp Rp · dimRp H 1 (OK,S , V ) = dimOp Rp · X dimRp H 1 (Kp , V ) v|p = X dimOp H 1 (Kp , V ) v|p Applying Tate’s Euler characteristic formula dimOp H 0 (Kv , V ) − dimOp H 1 (Kv , V ) + dimOp H 2 (Kv , V ) = − [Kv : Qp ] dimOp , with H 0 (Kv , V ) = H 2 (Kv , V ) = 0, we obtain dimOp Rp · dimRp H 1 (OK,S , V ) =[Kp : Qp ] dimOp V =[Kp : Qp ] dimOp Rp , where the last equality follows from dimRp V = 1. 64 Further, we find dimOp Rp · dimRp H 1 (OK,S , V ) =[Kp : Qp ] dimOp Rp , =[Kp : Qp ] dimOp B \ = dimOp B(K p ) ⊗Zp Qp , \ ⇒ dimRp H 1 (OK,S , V ) = dimRp B(K p ) ⊗Zp Qp , so that H 1 (OK,S , V ) has dimension 1 over Rp . The dual exponential map (see [13] Chapter II §1.2-§1.4) is exp∗ : H 1 (K, V ) → H 0 (B, Ω1B/K ), and gives us a map det(exp∗ ⊗ id) : ∆Rp (OK,S , V ) → Ξ ⊗ Qp , Rp the determinant of Kato’s exp∗ ⊗ id. Note that for v ∈ S − {∞}, all terms in RΓf (Kv , V ) are zero except for possibly Hf1 (Kv , V ), and that all terms in RΓ(C, V ) are zero except for possibly H 0 (C, V ). The modules involved in the refined Burns–Flach Conjecture are in fact isomorphic to those just defined above. We summarize the situation in the 65 following two lemmata. Lemma 6.0.3. det RΓc (OK,S , T )−1 ≃ ∆Rp (OK,S , T ) Rp f c (OK,S , T ) which is defined in this case by Proof. We use the complex RΓ the exact triangle M f c (OK,S , T ) → RΓ(OK,S , T ) → RΓ RΓ(Kv , T ). v∈S−{∞} We consider the following exact triangle which it satisfies: RΓc (OK,S , T ) / RΓ f (O c K,S , T ) / RΓ(C, T ) Tate–Poitou duality gives f c (OK,S , T ) ≃ det RΓ(OK,S , T ∗ (1))∗ [−3] det RΓ Rp Rp (6.1) 66 So from (6.1), f c (OK,S , T ) ⊗Rp det RΓ(OK,S ⊗ R, T )−1 det RΓc (OK,S , T (1)) ≃ det RΓ Rp Rp Rp ≃ det RΓc (OK,S , T ∗ (1))∗ [−3] ⊗Rp det RΓ(OK,S ⊗ R, T )−1 Rp Rp −1 ≃ det RΓc (OK,S , T ∗ (1))∗ ⊗Rp det RΓ(OK,S ⊗ R, T )−1 Rp Rp ≃ det RΓc (OK,S , T ∗ (1))# ⊗Rp det RΓ(OK,S ⊗ R, T (−1)) Rp Rp since T ∗ (1) ≃ T # , whence T ∗# ≃ T (−1), ≃ det RΓc (OK,S , T ) ⊗Rp det RΓ(OK,S ⊗ R, T (−1)) Rp Rp where “∗” denotes RHom(•, Zp ) and “#” means changing the Rp -action via the (Rosati) involution on Rp . We conclude that det RΓc (OK,S , T )−1 Rp  −1  −1 = det(RΓ(OK,S , T )) ⊗Rp det(RΓ(OK,S ⊗ R, T (−1))) Rp Rp is isomorphic to ∆Rp (OK,S , T ). This proves Lemma 6.0.3. Lemma 6.0.4. The diagram detRp RΓc (OK,S , V )−1 ϑp  / Ξ ⊗ Qp Lemma 6.0.3  ∆Rp (OK,S , V ) detRp (exp∗ ⊗id) / Ξ ⊗ Qp 67 commutes, where det(exp∗ ⊗ id) Rp the determinant of Kato’s exp∗ ⊗ id. Proof. The map ϑp is constructed from the dual exponential maps Hf1 (Kv , Vp (B))∨ exp∗ / H 0 (B, Ω1B/K ), ≃ for v | p. The left vertical arrow is obtained from dualities and defining exact triangles, so that the composition of the left vertical map and the lower horizontal map detRp (exp∗ ⊗ id) is another exponential map, which at v | p is exp∗ . Knowing that the last two results are true, we can now pose the refined Burns–Flach Conjecture in the terminology of Kato. Conjecture 6.0.5. We have ϑ−1 ∞ (R L(B, 1)) ⊂ Ξ ⊗ 1 (6.2) Rp · det(exp∗ ⊗id)−1 (ϑ−1 ∞ (R L(B, 1)) = ∆Rp (OK,S , T ). (6.3) and 68 Chapter 7 Proof of the conjecture. The proof will be indicated in this chapter, with more details being given in the following chapters. 7.1 A universal situation. Following Kato, we now pass to a situation universal with respect to invertible sheaves over p-adic rings such as Tp (B) over Rp . Write K m for the ray class field modulo m, and let f be the conductor of B. Recall that S is the set of places of K dividing ∞fp. Put n Λn :=Zp [Gal(K fp /K)], ∞ fp Λ∞ :=lim ←−Λn = Zp [Gal(K /K)] (the completed Galois group algebra), n Fn :=(fK fpn )∗ (fK fpn )∗ (Zp (1)SpecOK,S ) (fL being the canonical morphism SpecOL,S → SpecOK,S ), F∞ :=lim ←−Fn n 69 We write ∆Λ∞ (OK,S , F) for the invertible Λ∞ -module  −1  −1 det(RΓ(OK,S , F)) ⊗Λ∞ det(RΓ(OK,S ⊗Z R, F(−1))) . Λ∞ Λ∞ The action of Gal(K/K) on Tp (B)(−1) is given by a character Gal(K/K) / / Gal(K fp∞ /K) / R× , p inducing a ring homomorphism Λ∞ → Rp , so that T ≃ F∞ ⊗Λ∞ Rp and ∆Λ∞ (OK,S , F∞ ) ⊗Λ∞ Rp ≃ ∆Rp (OK,S , T ). To better describe ∆Λ∞ (OK,S , F∞ ), we calculate the cohomology of OK,S with F∞ -coefficients: Lemma 7.1.1. • H 0 (OK,S , F∞ ) = 0 × • H 1 (OK,S , F∞ ) = ← lim −OK fpn ,S ⊗ Zp n • There is a natural short exact sequence sum=0  n lim ←−PicOK fp ,S ⊗Z Zp / n / H 2 (OK,S , F∞ ) / / lim  M n w|finite v∈S ←− Zp  . 70 Proof. We use the short exact sequence / µpk 0 pk /G m /G /0, m for which the Kummer sequence yields the following exact sequences: 0 / H 0 (O / / H 0 (O H 0 (OK fpn ,S , Gm )/pk / / H 1 (O / / H 1 (O H 1 (OK fpn ,S , Gm )/pk / H 2 (O / H 2 (O K fpn ,S , µpk ) K fpn ,S , Gm )pk K fpn ,S , µpk ) K fpn ,S , Gm )pk K fpn ,S , µpk ) K fpn ,S , Gm )pk BrS (OK fpn )pk Take projective limits ← lim − to obtain the following exact sequences: k 0 / H 0 (O /0 OK fpn ,S ⊗Z Zp , / / H 1 (O //0 K fpn ,S , Zp (1)) K fpn ,S , Zp (1)) sum=0  PicOK fpn ,S ⊗Z Zp / / H 2 (O K fpn ,S , Zp (1)) // M w|finite v∈S Take projective limits again, this time ← lim −, we find n H 0 (OK,S , F∞ ) = 0 Zp  71 and 1 n lim ←−OK fp ,S ⊗Z Zp = H (OK,S , F∞ ), n and obtain another exact sequence: sum=0  n lim ←−PicOK fp ,S ⊗Z Zp / n / H 2 (OK,S , F∞ ) / / lim  M ←− n Zp  w|finite v∈S Thus the result is proved. We now use a construction of Kato ([13], III, §1.2). We will use his Assumption 7.1.2. K has class number 1. Kato’s E, and its Grössencharacter ν, we will denote by E0 and ϕ, respectively. Now, our E0 /K is an elliptic curve isogenous to our E of Chapter 5 and Chapter 6, and with complex multiplication by the full ring of integers OK . Then our “ϕ” here is consistent with that of those chapters. We use the following proposition of Kato (written using our notation). Proposition 7.1.3 ([13], III, 1.1.5). Suppose a ∈ End(E0χ ), (a, 6) = 1. Then there exists a unique rational function θa ∈ K(E0χ )× satisfying the following two conditions: Nb (θa ) = θa for any b ∈ End(E0χ ) such that (a, b) = 1. (7.1) The divisor of θa is deg(a)(e) − ker(a). (7.2) 72 Choose a non-zero ideal a of OK prime to 6fp, and such that N(a)ψ(a)−1 is not a root of 1. Set a = ψ(a). Let n Υa,n : H 0 (K fp ⊗K R, Zp ) ≃ M H 0 (C, Zp ) → H 1 (OK fpn ,S , Zp (1)) ι be the homomorphism (aι )ι 7→ X aι (ι−1 (θa (exp(hn ))), ι n where ι ranges over all embeddings K fp → C whose restriction to K concides n with the given embedding of K into C (i.e., over all finite places of K fp ), and where hn = π −n g −1 h, π = ψ(p), g a generator of f. Now hn will be an OK -basis of (fpn )−1 H1 (E0 (C), Z). (And θa (exp(hn )) is independent of the choice of h and g because θa ◦ [u] = θa for any u ∈ (OK )× .) Then 0 1 lim ←−Υa,n : H (C, F∞ (−1)) → H (OK,S , F∞ ) n corresponds to an element of H 1 (OK,S , F∞ ) ⊗Λ∞ H 0 (C, F∞ (−1))−1 , which we denote by za,Λ∞ (OK,S , F∞ ). Define za,Rp (OK,S , T ) ∈ H 1 (OK,S , T ) ⊗Rp H 0 (K ⊗ R, T (−1))−1 73 to be the image of za,Λ∞ (OK,S , F∞ ) under the map H 1 (OK,S , F∞ ) ⊗Λ∞ H 0 (K ⊗ R, F∞ (−1))−1  H 1 (OK,S , T ) ⊗Rp H 0 (K ⊗ R, T (−1))−1 . Define zRp (OK,S , T ) ∈ H 1 (OK,S , T ) ⊗Rp H 0 (C, T (−1))−1 to be N(a) − N(a)χT (σa )−1
−1 za,Rp (OK,S , T ), where χT is the action on T . Here N(a) − N(a)χT (σa ) is invertible in Rp because χT (σa ) = N(a)ψ(a)−1 α for a root α of 1 in K[G]. Then zRp (OK,S , T ) is independent of the choice of a. Define zΛ∞ (OK,S , F∞ ) ∈ H 1 (OK,S , F∞ ) ⊗Λ∞ H 0 (C, F∞ (−1))−1 ⊗Λ∞ quot(Λ∞ ) to be (N(a) − σa )−1 za,Λ∞ (OK,S , F∞ ). Observe that we have constructed elements za,Λ,∞ (OK,S , F∞ ) l lll lll l l l ulll za,Rp (OK,S , T ) SSSSS SSS SSS SSS ) RRRR RRR RRR RRR ) zRp (OK,S , T ) zΛ∞ (OK,S , F∞ ). 74 7.2 The strategy of proof. The rest of our exposition has two parts. First, following the approach of Kato, we show in §8 that zΛ∞ (OK,S , F∞ ) is an Λ∞ -basis of ∆Λ∞ (OK,S , F∞ ), which allows us to conclude that zRp (OK,S , T ) is a Rp -basis of ∆Rp (OK,S , T ). Second, we find (Theorem 9.0.2), using [13], III, Theorem 1.2.6, that the image of zRp (OK,S , T ) under the map det(exp∗ ⊗ id) : det ∆Rp (OK,S , V ) → Ξ ⊗ Qp Rp Rp is sent by ϑ∞ to (LS ((ϕχ)−1 , 0))χ∈Ĝ , where LS ((ϕχ)−1 , 0) equals the L-value L(ϕχ, 1), deprived of the Euler factors at primes dividing f. But the Euler factors at primes of bad reduction are 1, and we shall assume that the Euler factors at any other primes dividing f, are also units at p. So we may disregard the missing Euler factors. Our result on the image of zRp (OK,S , T ) shows that the image of det ∆Rp (OK,S , T ) Rp under exp∗ is generated by (ϑ∞ )−1 (R L(B/K, 1)), as asserted by the Burns– Flach Conjecture. We summarize the strategy of proof, including the translation of the re- 75 fined Burns–Flach Conjecture into the terminology of Kato, in the following diagram: detRp RΓc (OK,S , V )−1  ≃ ϑp / Ξ ⊗ Qp ≃ ϑ∞ /R⊗R ≃ detRp (exp∗ ⊗id) / Ξ ⊗ Qp ≃ ϑ∞ /R⊗R Lemma 6.0.3  ∈ ∈ ∆Rp (OK,S , T ) ∈ ⊂ ∆Rp (OK,S , V ) zRp (OK,S , T )  / +  [χ] L(ϕχ, 1)c[χ] P / (L(ϕχ, 1))[χ] The first line depicts the morphisms involved in the refined Burns–Flach Conjecture. The second line shows the translation into the terminology of Kato. On the third line is given the integral submodule of the vector space just above. On the fourth line is a basis for the integral submodule (on the left), and its images under the two maps of the second line. The element c+ [χ] is defined to be the pre-image, under ϑ∞ , of the element of R with [χ]-component 1 and all other components 0. 76 Chapter 8 A Λ∞-basis of ∆Λ∞ (OK,S , F∞). We will make an Assumption 8.0.1. • K has class number 1. • p 6 | [K(f0 p) : K], where f0 | f is the prime-to-p part of f. • p 6 | µ(HK ), where HK is the Hilbert Class Field of K. In this chapter we prove the following Theorem 8.0.2. Recall that the finite places of S are exactly those dividing fp. Then zΛ∞ (OK,S , F∞ ) is an Λ∞ -basis of ∆Λ∞ (OK,S , F∞ ). To work more easily with the element zΛ∞ (OK,S , F∞ ) ∈ H 1 (OK,S , F∞ ) ⊗Rp H 0 (K ⊗ R, F∞ (−1))−1 , 77 we fix a basis yΛ∞ (OK,S , F∞ ) of H 0 (K ⊗ R, F∞ (−1))−1 . Here yΛ∞ (OK,S , F∞ ) is an inverse system of embeddings n K fp ,→ C which could come, for example, from a fixed embedding Q ,→ C. Now also fix xa,Λ∞ (OK,S , F∞ ) ∈ H 1 (OK,S , F∞ ) such that za,Λ∞ (OK,S , F∞ ) = xa,Λ∞ (OK,S , F∞ ) ⊗Rp yΛ∞ (OK,S , F∞ ) and yΛ∞ (OK,S , F∞ ) is a basis of H 0 (K ⊗ R, F∞ (−1))−1 . Correspondingly, we express zΛ∞ (OK,S , F∞ ) = xΛ∞ (OK,S , F∞ ) ⊗Rp yΛ∞ (OK,S , F∞ ), where xΛ∞ (OK,S , F∞ ) = (N(a) − σa )−1 xa,Λ∞ (OK,S , F∞ ). 78 The primary result we will be using is the Theorem 8.0.3 ([17], Theorem 4.1: Main Conjecture). char(A∞ ) = char(U∞ /C∞ ), where A∞ =lim ←−An n U∞ =lim ←−Un n C∞ =lim ←−Cn , n n where An is the p-part of the ideal class group of K fp , Un is the global units, and Cn is the elliptic units. The inverse limits are all with respect to the norm maps. We need a lemma to permit us to deal with modules over discrete valuation ring, instead of with the Λ∞ -modules themselves. Lemma 8.0.4. Let N be a local Noetherian integral domain and suppose I, J ⊂ M are two invertible N -submodules of an invertible quotN -submodule M . (Since N and quotN are local, “invertible” means free of rank 1.) If N is Cohen-Macaulay then I = J if and only if Iq = Jq for all height-1 prime ideals q of N . 79 Proof. The result would follow if we could show that I= \ \ Iq ⊂ M and J = Jq ⊂ M. htq=1 htq=1 Since Iq = Nq ⊗N I and Jq = Nq ⊗N J, then this would in turn be implied by N= \ Nq ⊂ quotN. htq=1 The inclusion N⊆ \ Nq htq=1 is obvious. To prove N⊇ \ Nq , htq=1 let quotN 3 x 6∈ N, y where x, y ∈ N . Then y 6∈ N ∗ , ⇒ (y) 6= N , so that y is a (1-term) N -regular sequence. By [15] §15 Lemma 4, we find that dim N/(y) = dim N − 1. But [15] §16 Theorem 31(i) implies that dim N/(y) = dim N − ht(y). We conclude that ht(y)=1, which means that y lies in some height-1 prime 80 ideal q of N . Therefore xy 6∈ Nq and in particular \ x Nq . 6∈ y htq=1 Note 8.0.5. The ring Λ∞ is ∞ Zp [Gal(K f0 p /K)][[Gal(K f0 p /K f0 p )]] ≃ Zp [Gal(K f0 p /K)][[x, y]], where f0 is the prime-to-p part of f. If p 6 | [K f0 p : K] then Zp [Gal(K f0 p /K)] is a product of discrete valuation rings; hence Λ∞ is regular. If p | [K f0 p : K] then Λ∞ is no longer regular. But the ring Λ∞ is a complete intersection and in particular is Cohen–Macaulay. We will use the Main Conjecture to analyze ∆Λ∞ (OK,S , F∞ ). The following lemma relates the cohomology of OK,S to Rubin’s Iwasawa modules, to which we will then be able to apply the Main Conjecture. • H 1 (OK,S , F∞ ) ≃ US,∞ Lemma 8.0.6. • 0 / AS,∞ / H 2 (OK,S , F∞ ) / XS,∞ / 0 (exact) Note 8.0.7. The Main Conjecture deals with modules U∞ and A∞ , whereas here we are using the respectively corresponding modules US,∞ and AS,∞ , where the finite primes above those of S are inverted. We have the exact sequence 0 0 → U∞ → US,∞ → YS,∞ → A∞ → AS,∞ → 0 81 of Λ∞ -modules, which gives rise to the following exact sequence of Λ∞ -modules 0 0 → U∞ /C∞ → US,∞ /C∞ → YS,∞ → A∞ → AS,∞ → 0. 0 Moreover, for any height-1 prime q we have char(YS,∞,q ) = 0. (To see this, 0 consider a place v of K. If v has infinite inertial degree, we have Y{w|v},∞ itself being 0 since the norm maps on units (the maps between the various components of the projective limit) correspond to multiplication by the inertial degrees on the corresponding valuations. On the other hand, if v = p, then 0 since there only finitely many places above p in K∞ , then Y{w|v},∞ is finitely generated over Zp , hence pseudonull.) Proof of Lemma 8.0.6. Since × lim ←−OK fpn ,S ⊗ Zp ≃ lim ←−US,n = US,∞ , n n n lim ←−PicOK fp ,S ⊗Z Zp ≃ lim ←−AS,n = AS,∞ , n n and 0 / XS,∞ / YS,∞   lim ←− n / Zp  M Zp  finite w|v∈S then Lemma 7.1.1 immediately yields the desired result. /0, 82 Another lemma relates the constructed element xa,Λ∞ (OK,S , F∞ ) ∈ H 1 (OK,S , F∞ ) to the Iwasawa modules: Lemma 8.0.8. For all height-1 prime ideals q of Λ∞ , we have lengthΛ∞ ,q (Λ∞,q /xΛ∞ (OK,S , F∞ ) · Λ∞,q ) + lengthΛ∞ ,q XS,∞,q = lengthΛ∞ ,q C∞,q . Proof of Lemma 8.0.8. The ideal q induces a character Gal(Kfp /K) = Gtor ∞ ⊂ Λ →Λq → Λq /qΛq ≃ Qp (ψ), (where the last is some finite extension of Qp ) the prime-to-p part of which we denote d | f0 . Now C∞,q is spanned by an elliptic unit xdp∞ , while xf0 p∞ = xΛ∞ (OK,S , F∞ ) (both denoted using notation of the form Θ(v; L, a) in [19], II, §2), satisfying the Euler relations  NK f0 /K d (xf0 p∞ ) =   Y  xdp∞ , (1 − Frob−1 p ) p|f0 ,6 |d 83 where X NK f0 /K d = σ ∈ Λ. σ∈Gal(K f0 /K d ) Localizing at q, the norm element NK f0 /K d becomes a unit in Λq and thus xf0 p∞ a multiple of xdp∞ . Moreover, the element Y (1 − Frob−1 p ) p|f0 ,6 |d is also the charactistic ideal of XS,∞,q . The result follows. Now we are ready to give the Proof of Theorem 8.0.2. We will consider the situation locally, one prime at a time. (We will use the fact that char(M ) = Y qlengthRq Mq htq=1 for any finitely generated torsion Λ∞ -module M .) We aim to prove that zΛ∞ (OK,S , F∞ ) is a (Λ∞ )q -basis of ∆Λ∞ (OK,S , F∞ )q , for any height-1 prime q. We will then conclude from Lemma 8.0.4 that zΛ∞ (OK,S , F∞ ) is a Λ∞ -basis of ∆Λ∞ (OK,S , F∞ ). Fix a height-1 prime q of Λ∞ . We may apply Lemma 8.0.4 here. 84 Considering the integral complex RΓ(OK,S , F∞ )q , the result will follow if we can show that H 1 (OK,S , F∞ )q /((Λ∞ )q · xΛ∞ (OK,S , F∞ )) and ≃ H 2 (OK,S , F∞ )q have the same length as Λ∞,q -modules. By the first assertion of Lemma 8.0.6, and 8.0.8, this last statement is equivalent to charΛ∞,q (US,∞,q /C∞,q ) · charΛ∞,q XS,∞,q 0 =charΛ∞,q H 2 (OK,S , F∞ )q · charΛ∞,q YS,∞,q , then by the second assertion of Lemma 8.0.6, to charΛ∞,q (US,∞,q /C∞,q ) · charΛ∞,q XS,∞,q 0 =charΛ∞,q XS,∞,q · charΛ∞,q AS,∞,q · charΛ∞,q YS,∞,q , 85 which is rephrased lengthΛ∞,q (US,∞,q /C∞,q ) 0 . =lengthΛ∞,q AS,∞,q + lengthΛ∞,q YS,∞,q In view of Note 8.0.7, the last statement becomes lengthΛ∞,q (U∞,q /C∞,q ) = lengthΛ∞,q A∞,q . Consequently, the statement to be proved follows from the Main Conjecture. 86 Chapter 9 The image of the basis. We wish to consider the [χ]-components of R individually, where [χ] is the orbit of χ ∈ Ĝ under AutK C. Accordingly, choose for each [χ] a pre-image c+ [χ] , under ϑ∞ in Ξ, of the element of R with [χ]-component 1 and the other components 0. We rephrase a result of Kato to identify the image of the basis zΛ (OK,S , T ). In his setting, it is assumed that the class number of K is 1, but that assumption is not actually used in the result we apply here. First we introduce Kato’s notation. Let A be a finite product of finite extensions of K. Let Λ = A ⊗ Qp . Let F be an invertible smooth Λ-sheaf on X = Spec(OK,S ) such that there exists an integer r ≥ 1 and a homomorphism λ : Gal(K ab,S /K) → A× which is continuous for the discrete topology of A× satisfying the condition that χF = χcyclo (σ)χE (σ)λ(σ)−r for all σ ∈ Gal(K ab,S /K), where χF : Gal(K ab,S /K) → (OK ⊗ Zp )× ⊂ A× is the action on F. Define Σ = H1 (B(C), Q)−1 and Ω = H 0 (B, Ω1B/K ). The result, in the notation of Kato, is: Theorem 9.0.1 ([13], Theorem III.1.2.6). The element zΛ (X, F) is sent 87 by the map exp∗ ⊗ id : H 1 (X, F) ⊗A Σ−1 → (Ω ⊗A Σ−1 ) ⊗ Qp to an element of Ω ⊗A Σ−1 whose image in A ⊗ R coincides with (−1)r−1 LA,S (X, F∗ (1), 0). In our case r = 1, F = T , A = R, and LA,S (X, F∗ (1), 0) = LS ((ϕχ−1 , 0). Kato’s result becomes: Theorem 9.0.2. The image of zΛ (OK,S , T ) under the map exp∗ ⊗id : H 1 (OK,S , T )⊗R H1 (B(C), Q) → H 0 (B, Ω1B/K )⊗R H1 (B(C), Q)⊗Qp , with respect to the chosen bases, has [χ]-component LS ((ϕχ)−1 , 0). We recall in this chapter the proof of this theorem. Fix, for the proof, an n embedding ι : K ab,S → C, and thus, by restriction, embeddings ιn : K fp → C. Denote by u := (un )n the norm-compatible system n fp × (ι−1 ) )n . n (θa (exp(hn )))n ∈ ((K Define the OK,p -basis ξ of H 0 (K ab,S , T ) to be the image of g −1 h under the isomorphisms H1 (E0χ (C), Z) ⊗ Zp ≃ H 0 (C, T ) ≃ H 0 (K ab,S , T ) 88 (where the right-hand isomorphism is induced by ι). Let ξn = π −n ξ mod T ∈ V /T . Then the Coleman power series gu,ξ associated to u and ξ is the function z 7→ θa (z + ι−1 (exp(g −1 h))) cχ . on E 0 n Fix n for the rest of the proof. Define s ∈ H 1 (K fp , T ∨ (1)) to be the image of u under n fp × lim ←−(K ) 1 / lim ←−H (K fpn , Zp (1)) m m n n  . ∪(ξm )∨ m 1 fp ∨ m 1 fp ∨ m lim ←−H (K , T (1)/π ) o trace lim ←−H (K , T (1)/π ) m m ≃ 1  H (K fpn , T ∨ (1)) We will shortly apply the Theorem 9.0.3 (Explicit Reciprocity Law). ([13], II, 2.1.7) The function n n exp∗ : H 1 (K fp , T ∨ (1)) → coLie(E0χ ) ⊗OK K fp sends s to π −n    d −n ω⊗ log(ϕ (gu,ξ )) (ξn ) ω (Here ω can be taken to be any OK -basis of coLie(E0χ ).) 89 Consider the composite map H 0 (B, Ω1B/K )∨ →H 1 (OK,S , T ) →H 1 (OK fpn ,S ), T −1 (1)) n →H 1 ((K fp )p , T −1 (1)), where the first arrow is defined by the element za,Λ (OK,S , T ). Consider the element of HB∨ whose v-component is h−1 v if v agrees with our embedding ι : K ab,S → C and is 0 if v does not. This element is sent to n g −1 · s ∈ H 1 (K fp Kp , T −1 (1)) by the above composite map. (We have related the Coleman power series gu,ξ to the element za,Λ (OK,S , T ).) Applying the Explicit Reciprocity Law, we find that this element is sent by exp∗ to g −1 ·π −n    d n −n log(ϕ (gu,ξ )) (ξn ) ∈ coLie(E0χ ) ⊗OK K fp , ω⊗ ω n where ξn denotes ξ restricted to K fp . To put this expression in useful form we apply a lemma. We use θ(ϕχ)(a) , as described by Proposition 7.1.3: Lemma 9.0.4. ([13], III, 1.1.6) Write τ =  n K fp /K a  n ∈ Gal(K fp /K), and let 90 n n α = exp(hn ) ∈ (E0χ )fpn (C) = (E0χ )fpn (K p ), σ ∈ Gal(K fp /K). The value of   d log(θ(ϕχ)(a) ) ω at σα is equal to −1 Z ω ! · N(a)Lσ-part ((ϕχ)−1 , 0) − (ϕχ)(a)Lστ -part ((ϕχ)−1 , 0) . hn Choose a ≡ 1( mod f), so that τ = 1 and the above expression is simplified. The image of za,Λ (OK,S , T ) we wish to ascertain is g −1 · π −n ω ⊗ −1 Z ω ! · N(a) − (ϕχ)(a) LS ((ϕχ)−1 , 0) hn It follows that the image of zΛ (OK,S , T ) = N(a) − N(a)χT (σa )−1
−1 za,Λ (OK,S , T ) is g −1 ·π −n −1 Z ω⊗ ω · LS ((ϕχ)−1 , 0) hn =g −1 ·π −n −1 Z ω⊗ ω · LS ((ϕχ)−1 , 0), π −n g −1 h −1 Z =ω ⊗ ω · LS ((ϕχ)−1 , 0), h −1 =c+ [χ] · LS ((ϕχ) , 0) =c+ [χ] · LS (ϕχ, 1). 91 This proves Theorem 9.0.2. 92 Chapter 10 Examples. In this chapter we will find examples, or show the existence of examples, of elliptic curves to which the our main theorem applies. We search for elliptic curves E/F , having complex multiplication by a nonmaximal order O = Z + f · OK , for which the Mordell-Weil group E(F ) has rank 0. We will make some simplifying assumptions: Assumption 10.0.1. • K has class number 1. • f is a prime ≥ 3. The field F contains the j-invariant of E. The ring class group is Gf := If , where If is the group of fractional ideals prime to f , Zf is the group of Zf Pf principal ideals with a generator in Q ⊂ K, and Pf is the group of principal ideals with a generator ≡ 1 mod f . The ring class field of O is a field Hf such that Gal(Hf /K) ≃ Gf . The theory of complex multiplication shows that we may take also: • F = Hf . 93 10.1 Elliptic curves with rank 0. Note 10.1.1. If E/F and E0 /F are two elliptic curves isogenous over F , then L(E/F, s) = L(E0 /F, s). We have the following theorem of Coates and Wiles ([5]), a generalization of a result of Arthaud: Theorem 10.1.2. If rank(E0 (F )) ≥ 1 then L(E0 /F, 1) = 0. Accordingly, we search for elliptic curves E0 for which L(E0 /F, 1) 6= 0. Then we will try to show the existence of elliptic curves E/F , isogenous over F , with complex multiplication by O = Z + f · OK . 10.2 Elliptic curves with complex multiplication by a nonmaximal order. Suppose E0 /K is an elliptic curve with complex multiplication by OK . There exists an elliptic curve E/K, with complex multiplication by O, fitting into the short exact sequence 0 /C / E0 /E /0, 94 with C finite. Theorem 10.2.1. E is defined over Hf . To prove this theorem we use the following Lemma 10.2.2. There exists a short exact sequence / Tf E0 0 / Tf E /C / 0. Proof of lemma. For each n ≥ 1, we have the diagram, with exact rows and columns: 0 0 /C  ·f n /C 0 0 E0,f n   / Ef n / C0  / E0  /E /0 ·f n ·f n   /E 0 /E 0 0  /0 n /0  The leftmost vertical map is 0 since C is f -torsion. By diagram-chasing, we get an isomorphism Cn0 → C. Chasing the diagram E0,f n  ·f E0,f n−1  0 / Ef n / C0 n /0 / C0 n−1 /0 ·f  / Ef n−1  0 95 0 shows that there is an isomorphism Cn0 → Cn−1 which is compatible with the 0 isomorphisms Cn0 → C and Cn−1 → C constructed above, in that the diagram ≃ Cn0 @ @@ @@ ≃ @@@  C / C0 n−1 zz z z zz ≃ z} z commutes. Then, taking the projective limit over diagrams / Ef n E0,f n  ·f  ·f / Ef n−1 E0,f n−1 / C0 n / C0  /0 n−1 / 0, /C / 0. we conclude that there is an exact sequence 0 / Tf E0 / Tf E Proof of Theorem 10.2.1. Consider the action of OK,f upon Tf E0 . It gives rise to a representation Zf [Gal(K/K)] → O× K,f . The image of Zf [Gal(K/Hf )] lies in (Zf + f OK,f )× . Therefore, Gal(K/Hf ) acts on Tf E, compatibly with 96 the Gal(K/Hf )-action on Tf E0 . As a result of the exact sequence 0 /C / E0 /0, /E Gal(K/Hf ) acts on C. The short exact sequence / Tf E0 0 / Tf E /C /0 from Lemma 10.2.2 illustrates that C is Gal(K/Hf )-stable. Consequently, we see from the previous exact sequence 0 /C / E0 /E /0 that E is also Gal(K/Hf )-stable. The result follows. Our strategy will be to find ring class fields Hf and elliptic curves E0 over F := Hf with L(E0 /F, 1) 6= 0. These will be examples of rank-0 elliptic curves, to which Gross’ Conjecture applies. Moreover, this will show the existence of rank-0 elliptic curves E with nonmaximal endomorphism ring, to which only the more general Burns–Flach Conjecture applies. By Theorem 10.2.1, we may as well search for E0 with complex multiplication by OK . We will in fact use curves E0 which are defined over K. Write, as before, ϕ for a Grössencharacter of K, which when pre-composed 97 with the norm, gives the associated Grössencharacter ψ of E0 over Hf . Then, as explained in [8], we have Y L(ψ, s) = L(ϕχ, s). \ χ∈Gal(H f /K) Since, by a result of Deuring for elliptic curves E0 with End(E0 ) = OK (see [21], II, Theorem 10.5(a)), we have L(E0 /F, s) = L(ψ, s)L(ψ, s), then we must verify exactly that \ L(ϕχ, 1) 6= 0 ∀χ ∈ Gal(H f /K). Let us examine the characters ϕ and χ more closely. We know, by the property of the associated Grössencharacter, that for x ≡ 1 mod c0 , where c0 is the conductor of E0 /K, ϕ((x)) = x. Since OK by assumption is a principal ideal domain, then ϕ is essentially the identity function on K × . Now χ may be considered a map Gf ≃ Gal(Hf /K) → C× 98 Lemma 10.2.3. Gf is a group of order 2Φ(f ) , φ(f )w where w = |O× K |, φ is Euler’s φ-function, and Φ is its obvious analog defined on ideals of OK . (A more precise description of this group will be given for examples we find.) Proof. First, Gf = (OK /f )× If ≃ . Zf Pf (Z/f )× im(O× K) Consider the archimedean absolute value. We observe that 1+f ·OK intersects the unit circle at only 1 point if f > 2, which is true here by assumption. × Therefore, the group homomorphism O× is an injection, and K → (OK /f ) × × |im(O× K )| = |OK | = w. Since there are two roots of unity in Z , the image of × ) K /f ) (Z/f )× in (O has order φ(f . Since 2 im(O× ) K (OK /f )× Φ(f ) = , × w im(OK ) the order of (OK /f )× (Z/f )× im(O× K) is 2Φ(f ) , φ(f )w as desired. 99 10.3 A first attempt. Here we examine a first and mostly unsuccessful attempt to find examples. We also explain why many of the curves of the class searched, can be ruled out by theory alone. Use K = Q(i), which has class number 1. If f ≡ 1 mod 4, then f is split 2 −1) = f −1 . If f ≡ 3 mod 4, then f is inert in K, and in K, and |Gf | = 2(f 4(f −1) 2 2 −1) |Gf | = 2(f = f +1 . 4(f −1) 2 Consider the Gauss curve y 2 = x3 − x, which we denote E0 . It has abelian torsion, so that the above considerations for ψ apply. If we knew cf , then we would be guaranteed the existence that L(ϕχ, 1) 6= 0 for all χ ∈ G of an elliptic curve E, defined over Hf , with complex multiplication by exactly Z + f · OK and rankZ E(Hf ) = 0. As a preliminary search, a necessary (though insufficient) condition was checked. The L-function satisfies the functional equation Λ(ϕχ, s) = T (ϕχ)Λ(ϕχ, 2 − s), where s Λ(ϕχ, s) = (DN(ϕχ)) 2 (2π)−s Γ(s)L(ϕχ, s), where N(ϕχ) ∈ N is the norm of the conductor c(ϕχ) (an ideal of OK ) and D is the discriminant of K. And T (ϕχ) is related to the root number W (ϕχ) by W (ϕχ) . T (ϕχ) = −i p N(ϕχ) 100 Since E0 is defined over Q, then ϕ is anticyclotomic, i.e., ϕ(n) = ϕ(n). The same is true of χ. On the ideals of Z, χ is trivial (since all ideals of Z represent the trivial class in the ring class group). The Grössencharacter ϕ satisfies ϕ((a)) = a for a ≡ 1 mod (1 + i)3 , so that for a ∈ Z, ϕ((a)) = a for a ≡ 1 mod 4 and ϕ((a)) = −a for a ≡ 3 mod 4. This means that the value of ϕ is real on the ideals of Z. Thus L(ϕχ, 1) = L(ϕχ, 1), which means that any root number W (ϕχ) being unequal to 1 will force the L-value to be 0. For each prime f between 3 and 250, the root number W (ϕχ) was checked numerically for all characters χ of Gf . W (ϕχ) was found to be -1 for at least one χ for each f 6= 3. This can in fact be proven for all f > 3: cf such Proposition 10.3.1. Let f be any prime > 3. Then there exists χ ∈ G that W (ϕχ) 6= 1. Proof. We shall restrict our attention to nontrivial χ. We have as in Miyake ([16], 3.3.11), the formula for the root number of ϕχ: W (ϕχ) = W (ϕ)W (χ) ϕ(cχ ) χ(cϕ ), |ϕ(cχ )| where cϕ and cχ are the respective conductors of ϕ and χ. Let us examine ϕ(cχ ) the factors W (ϕ), W (χ), |ϕ(c , and χ(cϕ ). We shall determine the first two, χ )| and show that the fourth can be chosen, depending on the third, making the whole product -1. 101 We know that W (ϕ) = 1 since the Gauss curve is a rank-0 elliptic curve. Second, let us examine W (χ). The character χ is a character of the Galois group of the extension F = Hf /K. Consider the representation Q ResQ K IndK χ formed by induction and then restriction. It is   0  χ(x) x 7→  . 0 χ(x) Now since xx ∈ Z for x ∈ OK , and since the image of Z in Gf is trivial, then χ(x)χ(x) = χ(xx) = 1, so that χ is anti-cyclotomic (i.e., χ(x) = χ(x)). It follows that the character associated to the representation Q ResQ K IndK χ 102 is   0  χ(x) Q tr(ResQ  K IndK χ) =tr  0 χ(x)   0  χ(x) =tr   0 χ(x) =χ(x) + χ(x) ∈R. One also checks that tr(IndQ K χ)(x) ∈ R for x ∈ GQ \GK . The work of Fröhlich and Queyrut ([7]) tells us that for a real character which comes from a real representation, the root number is 1. Though the representation IndQ K χ is not itself real, it is conjugate to a real representation. Thus its root number is also 1. Since the L-function is unchanged by induction or restriction of characters, we see that W (χ) must be 1. ϕ(cχ ) Next, we examine the factor |ϕ(c . Since the conductor of χ is f if χ is χ )| ϕ(cχ ) nontrivial, then ϕ(cχ ) ∈ Z, and the factor |ϕ(c is ±1 . χ )| The class of cϕ = (1 + i)3 is of order exactly 2 in Gf since its square is the class of the ideal (8), which is trivial, while the class of (1 + i)3 is itself nontrivial. Therefore, there exists a character χ of Gf which has value −1 at cϕ . Since f > 3, then Gf 6=< 1 + i >, so that there is a nontrivial character of Gf which factors through Gf / < 1 + i >. 103 ϕ(cχ ) cf for which the product Whether |ϕ(c is 1 or -1, we find a χ ∈ G χ )| ϕ(cχ ) χ(cϕ ), |ϕ(cχ )| and thus the product ϕ(cχ ) χ(cϕ )W (ϕ)W (χ), |ϕ(cχ )| cf such that W (ϕχ) = −1. is -1. Thus, for f > 3, there exists χ ∈ G The curve E0 /H3 turned out to have non-zero L-value, and so E3 was the lone example obtained from this search. 10.4 A second search. A different class of curves was searched, yielding 15 elliptic curves over ring class fields F := Hf with L(E0 /F, 1) 6= 0. These are examples of rank-0 elliptic curves, to which Gross’ Conjecture applies. Moreover, this shows the existence of 15 rank-0 elliptic curves E with nonmaximal endomorphism ring, to which only the more general Burns–Flach Conjecture applies. Here the Gauss curve E0 is replaced by a quartic twist thereof. Let E4,0 be given by the equation y 2 = x3 − (1 + 4i)x. 104 √ This curve is isomorphic to the Gauss curve over F (4 1 + 4i), using the co√ ordinate change (x, y) = (β 2 x0 , β 3 y 0 ), where β = 4 1 + 4i. Since the Gauss curve y 2 = x3 − x has abelian torsion, then E4,0 does also. If we knew that cf , where ϕ0 is the Grössencharacter of E4,f , then we L(ϕ0 χ, 1) 6= 0 for all χ ∈ G would be guaranteed the existence of an elliptic curve E4,f , defined over Hf , with complex multiplication by exactly Z + f · OK . The Grössencharacter of E4,0 is ϕ0 = ϕ, where  is the character associated to the quartic twist E0 7→ E4,0 . We have 0  ϕ ((α)) = 1 + 4i α  α, α≡1 mod (1 + i)3 . 4 (See [12], Chapter 19, Theorem 5.) The conductor of  is 1 + 4i, which is prime to the conductor of ϕ, and to the conductor of χ if f 6= 17. In this case, the root number of ξ = ϕ0 χ can be calculated using the formula W (ϕ0 χ) = ϕ0 (cχ ) (cϕ0 )W (ϕ0 )W (χ), |ϕ0 (cχ )| where the root number of ϕ0 = ϕ is calculated using the formula W (ϕ) = ϕ(c ) (cϕ )W (ϕ)W (). |ϕ(c )| The values of L(ϕ0 χ, 1) were checked for all ring class characters χ, for all primes f between 3 and 61, excluding 17. The values L(ϕ0 χ, 1) were found all to be non-zero, except when f = 5 and χ was the non-trivial character of the two-element group G5 . In this case, L(ϕ0 χ, 1) appeared to be zero, though 105 this could not be deduced from the numerical calculations. In any case, it was determined that E4,0 /Hf had non-zero L-value for the following 15 values of f: 3, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61. Thus there exists an example E4,f for each of these values of f . 10.5 The code. Here we give the code used in PARI to find the examples. With some modification, many other similar classes of elliptic curves could be searched. Explanation is given interspersed with the code. Comments generally follow the line or lines to which they refer. bnf=bnfinit(y^2+1); Initializes the number field ‘bnf’= Q(i). bnr2=bnrinit(bnf,[1,4]~,1); Initializes ‘bnr2’, which is Q(i) with the structure of the ray class group modulo 1 + 4i. initialize(f)= Initializes the data involving f =‘eff’. { eff=f; 106 Sets f , called ‘eff’. bnr=bnrinit(bnf,eff,1); Initializes ‘bnr’, which is Q(i) with the structure of the ray class group modulo f. bnr3=bnrinit(bnf,eff*[1,4]~,1); Initializes ‘bnr3’, which is Q(i) with the structure of the ray class group modulo f (1 + 4i). prim=znprimroot(eff); for(index=1,eff, if(prim==Mod(index,eff), prim=index ) ); prim=bnrisprincipal(bnr,prim)[1]; Sets a generator ‘prim’ of (Z/f Z)∗ . chi=vector(length(bnr.clgp[2])); Sets χ to be the trivial character. } h1generator(bnf,ideal)= Extracts a generator of ‘ideal’ in ‘bnr’, assuming the class number of ‘bnr’ is 1. { dec=factor(idealnorm(bnf,ideal)); 107 gen=1; for(index=1,matsize(dec)[1], fac1=[dec[index,1],0]~; fac2=[1,0]~; for(a=1,sqrt(dec[index,1]), for(b=1,a, if(a^2+b^2==dec[index,1], fac1=[a,b]~; fac2=[a,-b]~; ) ) ); gen=nfeltmul(bnf,gen, nfeltpow(bnf,fac1,idealval(bnf, ideal,idealfactor(bnf,fac1)[1,1])) ); if(fac1!=[1,1]~& fac2!=[1,0]~, gen=nfeltmul(bnf,gen, nfeltpow(bnf,fac2,idealval(bnf, ideal,idealfactor(bnf,fac2)[1,1])) ); ); ); gen 108 } phieval(ideal)= Evaluates the character ϕ =‘phi’ (factor of the Grössencharacter) at ‘ideal’, given by a generator, expressed as a column vector on the integral basis. { for(o=0,3, if(nfeltmod(bnf, nfeltmul(bnf, ideal,nfeltpow(bnf,[0,1]~,o))-[1,0]~, nfeltpow(bnf,[1,1]~,3)) ==[0,0]~, ideal=nfeltmul(bnf, ideal,nfeltpow(bnf,[0,1]~,o)); ); ); if(nfeltmod(bnf,ideal,[1,1]~)==[0,0]~, ideal=[0,0]~; ); ideal[1]+ideal[2]*I } epsiloneval(ideal)= Evaluates the character  =‘epsilon’, associated to the quartic twist of E by 1 + 4i, at ‘ideal’. We know the character is either [1] or [3] as a character of the cyclic 4-group ‘bnr2.clgp’, and, twisting three times if necessary, we may 109 assume without loss of generality that it is [1]. { if(idealnorm(bnr2,idealcoprime(bnr2,[1,4]~,ideal))!=1, 0, exp(2*Pi*I*bnrisprincipal(bnr2,ideal)[1][1] *1/bnr2.clgp[2][1]) ) } rnepsilon=bnrrootnumber(bnr2,[1]); Calculates the (fixed) root number of  =‘epsilon’. phiprimeeval(ideal)= Evaluates ϕ0 =‘phiprime’= ϕ at ‘ideal’. { phieval(ideal)*epsiloneval(ideal) } cophiprime=nfeltmul(bnf,[1,4]~,nfeltpow(bnf,[1,1]~,3)); Calculates the conductor of ϕ0 =‘phiprime’= ϕ. rnphiprime=(1+4*I)/sqrt(17) *epsiloneval(nfeltpow(bnf,[1,1]~,3)) *1*rnepsilon; 110 Calculates the (fixed) root number of ‘phiprime’=‘phi’·‘epsilon’. (At 1+4i, the conductor of ‘epsilon’, ‘phi’ takes the value 1+4i.) chieval(ideal)= Evaluates the character χ =‘chi’ of the ray class group modulo f , at ideal. { if(idealnorm(bnr,bnrconductorofchar(bnr,chi))==1, 1, if(idealnorm(bnr,idealcoprime(bnr, bnrconductorofchar(bnr,chi),ideal))!=1, 0, exp(2*Pi*I*( sum(index=1,length(bnr.clgp[2]), bnrisprincipal(bnr,ideal)[1][index]*chi[index] /bnr.clgp[2][index] ) )) ) ) } xieval(ideal)= Evaluates the character ξ =‘xi’= ϕ0 χ = ϕχ at the ideal generated by ‘ideal’. { 111 phiprimeeval(ideal)*chieval(ideal) } rnxi()= This calculates the root number (varying with χ) of ξ =‘xi’= φ0 χ = φχ. { cochi=h1generator(bnr,bnrconductorofchar(bnr,chi)); fact=phiprimeeval(cochi); if(fact==0,0, fact/sqrt(norm(fact)) *chieval(cophiprime) *rnphiprime *bnrrootnumber(bnr,chi) ) } isring(upsilon)= Tests whether ‘upsilon’, a character of the ray class group modulo f (‘bnr.clgp’), factors through the ring class group modulo f , and thus is a valid χ to consider. Uses the constant ‘prim’, which is a generator of (Z/f Z)∗ , expressed as an element of ‘bnr.clgp’. { if(length(bnr.clgp[2])==1, upsilon[1]*prim[1]%(bnr.clgp[2][1])==0, 112 (upsilon[1]*prim[1]*bnr.clgp[2][2] +upsilon[2]*prim[2]*bnr.clgp[2][1]) %(bnr.clgp[2][1]*bnr.clgp[2][2])==0; Tests whether ‘upsilon’ factors through the quotient by (Z/f Z)∗ ,i.e., whether it gives character on the ring class group as well as the ray class group. ) } getapbp(p)= Sets ap =‘ap’ and bp =‘bp’ for the modular form. { if(idealprimedec(bnr,p)[1][4]==2, ap=0; bp=-xieval([p,0]~); ); if(idealprimedec(bnr,p)[1][3]==2, for(i=1,sqrt(p), for(j=1,i, if(i^2+j^2==p, ap=xieval([i,j]~); bp=0 ); ); ); 113 ); if(length(idealprimedec(bnr,p))==2, for(i=1,sqrt(p), for(j=1,i, if(i^2+j^2==p, v1=xieval([i,j]~); v2=xieval([i,-j]~); ap=(v1+v2); bp=(v1*v2) ); ); ); ); } add(n,P,y,z)= This function calls itself, and is the heart of the Gross–Buhler recursion. { Sum=Sum+y*expq^n/n; forprime(p=2,min(P,bound/n), getapbp(p); x=y*ap; if(p==P,x=x-z*bp); add(p*n,p,x,y); 114 ) } GB(bound)= { f1=expq; a1=1; Sum=a1*f1; forprime(p=2,bound, getapbp(p); add(p,p,ap,1); ); Sum } The preceding two functions comprise the algorithm of Gross and Buhler, as described in [1]. This algorithm can be described as “computationally minimal” in the sense that, using the known relations between the various an and bn (surely the most efficient way to calculate the terms), the minimum number of them necessary for the rest to be calculated, are kept in memory at once. Lval(xx,bound)= 115 This function approximates the L-value L(ξ, 1), using the formula L(ξ, 1) = ∞ ∞ X X an − 2πn an − 2πnx √ √ ne x N , e N + T (ξ) n n n=1 n=1 where N = |D|N(ξ), D being the discriminant of K. The computaion uses the cutoff point x =‘xx’ ≈ 1, and ‘bound’ terms of each series. Varying x does not change the infinite sum, of course, and this was used as a way to check the correctness of the calculation. { expq=exp(-2*Pi/sqrt(4*2^3*17*idealnorm(bnf, bnrconductorofchar(bnr,chi)))*xx); term1=GB(bound); expq=exp(-2*Pi/sqrt(4*2^3*17*idealnorm(bnf, bnrconductorofchar(bnr,chi)))*(1/xx)); term2=conj(GB(bound)); term1+rnxi*term2 } multichi1()= Checks the L-value for fixed f and all possible χ, for cyclic ray class group modulo f . Note that in all cases the ring class group modulo f is cyclic, but it is more convenient here to examine the characters of the ray class group, finding the L-value only for those which factor through the ring class group. (In some cases, the ray class group is not cyclic.) 116 { indicator=1; for(a=0,bnr.clgp[1]-1, if((isring([a])==1)&(indicator==1), chi=[a]; if(truncate(norm(1000*Lval(1,500)))==0, indicator=0 ); ); ); if(indicator==1, print("GOOD f=" eff ), print("BAD f=" eff) ) Prints “GOOD” if Ef is an elliptic curve of the desired type, and “BAD” otherwise. } multichi2()= Checks the L-value for fixed f and all possible χ, for ray class group modulo f with 2 cyclic components. Again, only those characters which factor through the ring class group are considered. { for(a=0,bnr.clgp[2][1]-1, 117 for(b=0,bnr.clgp[2][2]-1, if((isring([a,b])==1)&(indicator==1), chi=[a,b]; if(truncate(norm(1000*Lval(1,500)))==0, indicator=0 ); ); ); ); if(indicator==1, print("GOOD f=" eff ), print("BAD f=" eff) ) Prints “GOOD” if Ef is an elliptic curve of the desired type, and “BAD” otherwise. } multieff(lower,upper)= Checks the L-value for all (prime) f from ‘lower’ to ‘upper’. Note that for all f ≡ 3 mod 4, f is inert, which implies that the ray class group modulo f is cyclic. For all f > 5, ≡ 1 mod 4, f is non-inert and the ray class group has two cyclic components. For f =5, the ray class group is cyclic. We do not wish to examine f = 2 or 17 because they are not prime to the conductors of φ and , respectively. Thus ranges [‘lower’,‘upper’] are chosen, when this code 118 is used, which exclude 2 and 17. { forprime(index=lower,upper, initialize(index); if(length(bnr.clgp[2])==1, multichi1, multichi2 ); ); } 119 Bibliography [1] Buhler, J. and Gross, B., “Arithmetic on elliptic curves with complex multiplication. II.” Invent. Math. 79 (1985), 223—251. [2] Burns, D. and Flach, M., “Motivic L-functions and Galois module structures.” Math. Ann. 305 (1996), 65—102. [3] Burns, D. and Flach, M., “On Galois structure invariants associated to Tate motives.” American Journal of Mathematics 120 (1998), 1343— 1397. [4] Burns, D. and Flach, M., “Tamagawa numbers for motives with (noncommutative) coefficients.” Documenta Math. 6 (2001), 501—570. [5] Coates, J. and Wiles, A., “On the Conjecture of Birch and SwinnertonDyer.” Invent. Math. 39 (1977), 223—251. [6] Fontaine, J.-M. and Perrin-Riou, B., “Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L.” (French) (Seattle, WA, 1991), 599—706, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994. 120 [7] Fröhlich A. and Queyrut, J., “On the functional equation of the Artin Lfunction for characters of real representations.” Invent. Math. 20 (1973), 125—138. [8] Goldstein, C. and Schappacher, N., “Séries d’Eisenstein et fonctions L de courbes elliptiques multiplication complexe.” (French) J. Reine Angew. Math. 327 (1981), 184—218. [9] Gross, B., “On the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication.” (Cambridge, Mass., 1981), 219— 236, Progr. Math., 26, Birkhäuser, Boston, Mass., 1982. [10] Gross, B., Arithmetic on elliptic curves with complex multiplication. Springer Lecture Notes Math. 776. [11] Grothendieck, A., “Modèles de Néron et monodromie.” Exposé IX, SGA 7I, Springer–Verlag, 1972. [12] Ireland, K. and Rosen, M,, A Classical Approach to Modern Number Theory. Springer GTM 84, 1982. [13] Kato, K., Lectures on the approach to Iwasawa theory for Hasse-Weil Lfunctions via BdR . I. (Trento, 1991), 50—163, Lecture Notes in Math., 1553, Springer, Berlin, 1993. [14] Knudsen, F. and Mumford, D., “The projectivity of the moduli space of stable curves I: Preliminaries on ‘det’ and ‘Div’.” Math. Scand 39 (1976) 19—55. 121 [15] Matsumura, H., Commutative Algebra. Benjamin Cummings, 1980. [16] Miyake, T., Modular Forms. Springer–Verlag, 1989. [17] Rubin, K., “The ‘main conjectures’ of Iwasawa theory for quadratic imaginary fields.” Invent. Math. 103 (1991), no. 1, 25—68. [18] Rubin, K., Euler Systems. Annals of Mathematics Studies, 147. Hermann Weyl Lectures. The Institute for Advanced Study. Princeton University Press, Princeton, NJ, 2000. [19] de Shalit, E., Iwasawa theory of elliptic curves with complex multiplication. Orlando: Academic Press (1987). [20] Silverman, J. H., The Arithmetic of Elliptic Curves. New York: SpringerVerlag (1986). [21] Silverman, J. H., Advanced Topics in the Arithmetic of Elliptic Curves. New York: Springer-Verlag (1994).