Four-Dimensional Galois Representations of Solvable Type and Automorphic Forms Citation Martin, Kimball Lee (2004) Four-Dimensional Galois Representations of Solvable Type and Automorphic Forms. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4W9M-GD42. https://resolver.caltech.edu/CaltechETD:etd-05272004-224316 Abstract We investigate the Artin and strong Artin conjectures for continuous, four-dimensional complex Galois representations of solvable type. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Artin's conjecture; Langlands correspondence; modularity Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Scott Russell Johnson Prize for Excellence in Graduate Study in Mathematics, 2003. Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2004. Thesis Availability: Public (worldwide access) Research Advisor(s): Ramakrishnan, Dinakar Thesis Committee: Ramakrishnan, Dinakar (chair) Wales, David B. Goins, Edray Kilford, Lloyd Flach, Matthias Defense Date: 26 May 2004 Record Number: CaltechETD:etd-05272004-224316 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-05272004-224316 DOI: 10.7907/4W9M-GD42 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 2141 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 01 Jun 2004 Last Modified: 21 Jan 2021 00:30 Thesis Files Preview PDF - Final Version See Usage Policy. 464kB Repository Staff Only: item control page

Four-dimensional Galois representations of solvable type and automorphic forms Thesis by Kimball Martin In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2004 (Defended May 26, 2004) ii c 2004 Kimball Martin All Rights Reserved iii Acknowledgements Foremost, I must thank my advisor, Dinakar Ramakrishnan, for the wisdom and guiding powers of Ganesh, the unending patience of Vishnu, and the editing powers of Shiva. I cannot say enough about him, so I think I will not try. Next, I wish to thank the others on my committee—Matthias Flach, Edray Goins, Lloyd Kilford, and David Wales—for their willingness to help and for their questions and advice. Many people have helped shape this thesis, some more directly than others. I would especially like to thank the following: Michael Aschbacher, for always being able to tell me anything I wanted to know about finite groups. Jim Cogdell, for reading through [Ma1] and [Ma2] and giving much helpful advice, education and inquiry. In particular, Section 6.3 was written in response to him. Wee Teck Gan, for a question that led to Appendix A. Edray Goins, for summarizing what is known about icosahedral representations. David Wales, for reading through several manuscripts in various stages, offering many comments, teaching me much about finite groups and representations and showing me the method in Chapter 5. The referee of [Ma1], for many good suggestions on that paper which have also been incorporated into the thesis. Also Guy Henniart, Henry Kim, and Freydoon Shahidi for informing me of various results, most notably [He2], [Ki2], and Section 10 of [KS]. I thank the following institutions for unburdening me: The California Institute of Technology, and especially the Department of Mathematics, for supporting me and hosting me for five wonderful years. The National Science Foundation, for providing me with a fellowship in the first three years of graduate studies. The Fields Institute, with support from the National Science Foundation, for an educational, productive and amazingly cold winter of 2003. Finally, I appreciate all my friends who made my time in graduate school enjoyable, if at times less productive. Unfortunately, most of them do not have names. David Whitehouse does. Joso does not. iv Abstract Let F be a number field and ρ : Gal(F /F ) → GLn (C) be a continuous, irreducible representation. Artin conjectured that if ρ is non-trivial, then the associated L-function L(s, ρ) is entire. Langlands generalized this conjecture by asserting that there should be a cuspidal automorphic representation π of GLn (AF ) such that L(s, ρ) and L(s, π) agree at almost all places. If such a π exists, ρ is said to be modular. Langlands’s conjecture does indeed imply Artin’s conjecture. We consider in the thesis the case where ρ is a four-dimensional representation of solvable type, i.e., the image of ρ is solvable. We study what is known about Artin’s and Langlands’s conjectures for ρ. Artin’s conjecture is already known in the imprimitive cases, but not in the primitive ones. We show in two new cases, one primitive and one monomial, that ρ is modular; the former case yields new instances of Artin’s conjecture. We show that there are only two other primitive cases where one does not know Langlands’s conjecture for ρ, and that these cases are symplectic and would follow from certain instances of non-normal quintic base change for GL4 . Our new monomial case is non-essentially-self-dual. In fact we show that if ρ is monomial and essentially self-dual, then it is modular. We have two other small results for representations in other dimensions. First, if ρ is primitive and three dimensional, then in certain cases we show that the associated eightdimensional representation Ad(ρ) is modular. Second, we show that ρ of dimension n having supersolvable image is modular if n = 2j or n = 2j · 3 for some j. Lastly, we include in an appendix a proof of Ramakrishnan that if ρ corresponds to π as above, then the complete L-functions for ρ and π are equal as Euler products over Q. Q Q More precisely, L(s, ρv ) = L(s, πv ) at every finite place, and v|∞ L(s, ρv ) = v|∞ L(s, πv ). 1 Contents Acknowledgements iii Abstract iv 1 Introduction 3 2 Preliminaries on Linear and Finite Groups 7 2.1 2.2 2.3 Linear Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Classical Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.2 Orthogonal and Symplectic Groups . . . . . . . . . . . . . . . . . . . 8 Finite-Dimensional Representations . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Self-Dual Representations . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Clifford’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 A Little Finite Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 The Fitting and Frattini Subgroups . . . . . . . . . . . . . . . . . . 12 2.3.2 Extraspecial Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Preliminaries on Galois and Automorphic Representations 14 3.1 Galois Representations and Artin’s Conjecture . . . . . . . . . . . . . . . . 14 3.2 Automorphic Representations and Langlands’s Modularity Conjecture . . . 18 3.3 Functoriality Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3.1 Base Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3.2 Symmetric and Exterior Powers . . . . . . . . . . . . . . . . . . . . . 25 3.3.3 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.4 Applications to the Modularity Conjecture . . . . . . . . . . . . . . 26 2 4 The Method 28 4.1 The Reducible Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 The Irreducible Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5 Solvable Primitive Subgroups of GL4 (C) 34 5.1 The Fitting Subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.2 The Structure of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6 The Primitive Cases 43 6.1 The Case E24 · C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2 The Cases E24 · D10 and E24 · F20 . . . . . . . . . . . . . . . . . . . . . . . . 45 6.3 Transfer to GSp4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7 Monomial Representations 51 7.1 The Structure of Im(ρ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.2 Hypertetrahedral Representations . . . . . . . . . . . . . . . . . . . . . . . . 56 8 Three-Dimensional Representations 60 8.1 Primitive Three-Dimensional Representations . . . . . . . . . . . . . . . . . 60 8.2 Ad : GL3 → GL8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 9 Supersolvable Representations 65 Appendix α 69 B GAP Notation 71 3 Chapter 1 Introduction Let F be a number field and let ρ be an irreducible continuous representation of the absolute Galois group GF of F into GLn (C). Recall that there is associated to ρ an Artin Lfunction L(s, ρ) which is meromorphic on the whole complex plane. When ρ is the trivial representation, L(s, ρ) = ζF (s) is the Dedekind zeta function of F , which has a simple pole at s = 1 and is analytic elsewhere. Artin conjectured that if ρ is not the trivial representation, then L(s, ρ) is entire. This conjecture was proved for n = 1 by associating continuous characters of GF to Hecke characters over F , which were known to have entire L-functions. Langlands generalized this idea and further conjectured that, for any n, ρ corresponds to a cuspidal automorphic representation π of GLn (AF ). If there exists such a π, then ρ is said to be modular. This conjecture is known as the strong Artin conjecture or Langlands’s modularity conjecture. Because L(s, π) is entire, the strong Artin conjecture does indeed imply the Artin conjecture. Examples of when Artin’s conjecture is known but the strong Artin conjecture is not will be given below. In Section 3.3.4, we summarize several results about these conjectures. Progress has primarily been made for representations of solvable type, i.e., when the image of ρ is solvable. Some cases where ρ of solvable type is known to be modular are: when n = 2 ([La2], [Tu]); when n = 3 and ρ is induced ([JPSS]); when n = 3 and ρ is the twist of a symmetric square ([GeJ]); or when n = 4 and the image of ρ is orthogonal ([Ra2]). Also in the non-solvable case, certain two-dimensional icosahedral representations are known to be modular (for example, see [BDST], [Ta], [Goi], [Bu], [KW], [JM], [BS]). We consider irreducible Galois representations ρ : GF → GL4 (C) of solvable type. The 4 main goal of the thesis is to classify such representations ρ by their projective images into three categories: (i) representations ρ which can be proved to be modular; (ii) representations ρ for which we can prove the Artin but not the strong Artin conjecture; (iii) representations ρ for which we cannot prove Artin’s conjecture. Modularity in case (iii) will follow if certain instances of functoriality can be proved. Let ρ denote the corresponding projective representation and G = Im(ρ). Then G is a finite subgroup of PGL4 (C). Artin’s conjecture is known for ρ of solvable type unless ρ is primitive, and for the imprimitive cases modularity is known unless ρ is induced by a character of a quartic extension K/F with no intermediate subfields. The classification for projective images of primitive four-dimensional representations is known. A consequence of this classification is that for primitive ρ with solvable image, ρ maps into either GO4 (C) or GSp4 (C). In the former case, ρ is known to be modular by [Ra2]. In the latter, there are three possibilities for G: E24 · C5 , E24 · D10 and E24 · F20 . Here A · B denotes a group extension of B by A (which in these cases are actually semidirect products), E24 denotes the elementary abelian group of order 24 , C5 is cyclic, D10 is dihedral and F20 is Frobenius of order 20. We give a proof of these facts in Chapter 5. Our first result is the following Theorem 6.1. Suppose ρ : GF → GSp4 (C) ⊆ GL4 (C) has projective image G = E24 · C5 . Then ρ is modular. This appeared in [Ma1]. Consequently, we obtain new instances of Artin’s conjecture. We also show that in the other two cases, the modularity of ρ would follow from certain instances of (solvable) non-normal quintic base change for GL4 . Thus the two cases where G ≃ E24 · D10 and G ≃ E24 · F20 are the only ones in category (iii) above. In Chapter 7, we consider the case where ρ is imprimitive (of solvable type still). Then ρ is either in category (i) or (ii). If ρ is induced from a normal extension, then it is modular. So it suffices to assume ρ is induced from a character of a non-normal quartic extension with no intermediate subfields. We give some criteria about the projective images of such 5 representations and show that if ρ is essentially self-dual, then it is modular. One possible non-essentially-self-dual case occurs when G is an extension of A4 by the Klein group V4 . In this case, we call ρ hypertetrahedral and we are able to prove the following Theorem 7.1. Let F be a number field and ρ a hypertetrahedral representation of GF . Then ρ is modular. There are infinitely many such representations with projective image V4 · A4 which are not essentially self-dual. This gives new examples of non-normal induction and new examples of modular representations. These results are also contained in [Ma2]. Let us say a few words about how we prove these specific cases of modularity. First we briefly recall the ideas of Langlands’s proof of the tetrahedral case. See Chapter 3 for definitions of additional notation. Let σ : GF → GL2 (C) be a tetrahedral representation. Then there is a normal cubic extension K/F such that σK is modular. Say σK ↔ Π. There are three representations π0 , π1 , π2 of GL2 (AF ) whose base change πi,K to K is Π. One of these should actually correspond to σ. There is a unique π = πi whose central character matches with the determinant of σ. Then one proves Sym2 (σ) ↔ Sym2 (π). This combined with the correspondence σK ↔ πK allows one to conclude that, at any unramified place v, either σv ↔ πv or σ(Frv ) ∈ A4 has order divisible by 6. But A4 has no elements of order 6, so in fact σ ↔ π. In the proofs of Theorems 6.1 and 7.1 we will similarly have an normal cyclic extension K/F , which will either be cubic or quintic, such that ρK is modular. Then there will be either three or five πi ’s such that ρK ↔ πi,K . Among these πi ’s there is a unique one, π, such that Λ2 (ρ) ↔ Λ2 (π). Then one concludes that at any unramified place v either ρv ↔ πv or ρ(Frv ) ∈ G has order divisible by 6 or 10. But our projective group does not have any elements of order 6 or 10, so indeed ρ ↔ π. Chapter 4 presents more formally our method of proof. We also remark that the cases G ≃ E24 · D10 and G ≃ E24 · F20 are analogous to Tunnell’s octahedral case. Tunnell matched up an octahedral Galois representation σ to an automorphic represenation π such that σK ↔ πK for two extensions K, one normal and quadratic, the other non-normal and cubic. The latter part relied on the non-normal cubic base change of Jacquet, Piatetski-Shapiro and Shalika [JPSS]. Then, using the fact that S4 has no elements of order 6, Tunnell concluded that σ ↔ π. 6 If ρ is such that G ≃ E24 · D10 or G ≃ E24 · F20 , then knowing certain instances of nonnormal quintic base change would allow us to demonstrate the existence of an automorphic representation π such that ρK ↔ πK for both a normal quadratic extension K/F and a non-normal quintic extension K/F . As G has no elements of order 10, we could conclude that ρ ↔ π. One can also ask what can be said if ρ is reducible (of solvable type). The interesting cases are when ρ = χ ⊕ τ , where χ is a character and τ is a primitive, irreducible three-dimensional representation. In Chapter 8.1, we recall the classification for such threedimensional representations τ . There are essentially three cases, all of which are nonessentially-self-dual. Neither the Artin nor the strong Artin conjecture are known for these τ , but we show in two of these cases that the associated eight-dimensional self-dual representation Ad(τ ) is modular. In Chapter 9, using a direct application of automorphic induction, we deduce that if ρ is of dimension 2j or 2j · 3 and ρ has supersolvable image, then ρ is modular. Chapter 2 reviews some definitions and results on groups and representations which we will need. Chapter 3 covers relevant material on Galois and automorphic representations. In Appendix α, we give a proof of a result related to the strong Artin conjecture: if L(s, ρv ) = L(s, πv ) at almost all places v, then L(s, ρv ) = L(s, πv ) for all finite places v and Q Q v|∞ L(s, ρv ) = v|∞ L(s, πv ). This result, shown to us by Ramakrishnan, is not (to our knowledge) in the current literature. Many elementary properties of specific finite groups were computed in the computer algebra package [GAP]. In Appendix B, we give the GAP notation for many of the finite groups appearing in the text. 7 Chapter 2 Preliminaries on Linear and Finite Groups 2.1 Linear Groups 2.1.1 Classical Terminology Definition 2.1. Let x ∈ GLn (C). If det x = 1, we will say x is unimodular. Moreover, any subgroup of GLn (C) comprising only unimodular elements will be called unimodular.1 The group of all unimodular elements of GLn (C) is denoted SLn (C). For a matrix group G, Diag(G) will denote the subgroup of diagonal matrices contained in G. Definition 2.2. Let G ≤ GLn (C). If G is isomorphic to a subgroup of GLr (C) × GLn−r (C) for some 1 ≤ r ≤ n − 1, then we will say G is reducible. Otherwise, G is said to be irreducible. Definition 2.3. Let G ≤ GLn (C) be irreducible. Suppose Cn decomposes into subspaces V1 ⊕ · · · ⊕ Vm , m > 1, such that G acts (transitively) on set of subspaces {Vi }. Then G is said to be imprimitive. Otherwise, G is primitive. Observe that the condition that G is irreducible automatically ensures that, if G acts on a set of subspaces {Vi } where Cn = V1 ⊕ · · · ⊕ Vm , G acts transitively of {Vi }. Let ρ be the standard representation of a finite group G ≤ GLn (C). Then G is irreducible (resp. primitive) if and only if ρ is irreducible (resp. primitive). 1 Note our usage of the term “unimodular group” differs from that in the study of topological groups, i.e., we do not mean that it has a bi-invariant Haar measure. 8 2.1.2 Orthogonal and Symplectic Groups Let V be an n-dimensional vector space over C and let f : V × V → C be a non-degenerate bilinear form on V . Recall that, after fixing a basis for V , every bilinear form f can be associated to a matrix B by the defining equation f (x, y) = t xBy. We say f is orthogonal if f is symmetric, i.e., f (x, y) = f (y, x) for all x, y ∈ V ; and we say f is symplectic if f is skew symmetric, i.e., f (x, y) = −f (y, x) for all x, y ∈ V . Let O(f ) be the automorphism group of this space (V, f ), i.e., O(f ) = {g ∈ GL(V ) : f (g(x), g(y)) = f (x, y) for all x, y ∈ V } . (2.1) Similarly we can define the similitude group GO(V, f ) by GO(f ) = {g ∈ GL(V ) : f (g(x), g(y)) = λ(g)f (x, y) for all x, y ∈ V and some λ(g) ∈ C} . (2.2) The orthogonal and symplectic (similitude) groups we consider are subgroups of GLn (C) which respectively preserve orthogonal and symplectic forms. Since all orthogonal forms are equivalent over C, and similarly for symplectic forms (which only occur when n is even), we will make an explicit choice of orthogonal and symplectic forms and define the orthogonal and symplectic groups in terms of corresponding matrices.  Definition 2.4. The orthogonal group On (C) = x ∈ GLn (C) : t xx = I . The orthogonal  similitude group GOn (C) = x ∈ GLn (C) : t xx = λI, λ ∈ C . Here our orthogonal form is just the usual one, which in matrix representation is just the identity. Consider the symplectic form given by the 2m × 2m matrix,  J = 0 Im −Im 0  . (2.3)  Definition 2.5. The symplectic group Sp2m (C) = x ∈ GL2m (C) : t xJx = J . The sym plectic similitude group GSp2m (C) = x ∈ GL2m (C) : t xJx = λJ, λ ∈ C . Note that if we had chosen different forms, the corresponding orthogonal and symplectic groups would just be conjugates of the ones we defined. 9 Definition 2.6. Let G ≤ GLn (C). We will say that G is orthogonal (resp., symplectic) if G is conjugate to a subgroup of On (C) (resp., Spn (C)), and that G is of orthogonal (resp., symplectic) type if G is conjugate to a subgroup of GOn (C) (resp., GSpn (C)). 2.2 Finite-Dimensional Representations By character, we will always mean a linear character, i.e., a one-dimensional representation, unless we explicitly say the character of a representation. Let G be a group and ρ : G → GLn (C) be a linear representation. Denote the image of ρ by Im(ρ). Let ρ be the composition of ρ with the canonical projection from GLn (C) to PGLn (C). ρ / GLn (C) HH HH HH ρ HH#  G HH PGLn (C) Then ρ is the projective representation associated to ρ and the image Im(ρ) of ρ is called the projective image of ρ. We will say ρ is of solvable type if the image of ρ is solvable. This is evidently equivalent to the image of ρ being solvable. Let H ⊆ G. Then ρH denotes the restriction of ρ to H. If σ is a representation of H, then IndG H σ denotes the induction of σ from H to G. 2.2.1 Self-Dual Representations Let h , i be a (non-degenerate) orthogonal or symplectic form on V = Cn . The contragredient of ρ with respect to h , i is the unique representation which satisfies hρ(g)v, ρ̌(g)wi = hv, wi ∀g ∈ G, ∀v, w ∈ V. (2.4) While the contragredient depends upon the form chosen, it is independent of the choice of form up to equivalence. For instance, ρ̌ is equivalent to the representation of G given by g 7→ t ρ(g −1 ), which is the contragredient of ρ with respect to the standard orthogonal form on Cn . Hence we will simply speak of the contragredient representation, which is well-defined up to equivalence. We say that ρ is self-dual if ρ̌ ≃ ρ and that ρ is essentially self-dual if ρ̌ ≃ ρ ⊗ χ for some 10 character χ of G. In particular, if ρ ≃ ρ̌ ⊗ χ, then there is an orthogonal or symplectic form h , i (depending on the isomorphism between ρ and ρ̌ ⊗ χ) such that hρ(g)v, ρ(g)wi = χ(g)hv, wi ∀g ∈ G, ∀v, w ∈ V. (2.5) In addition, ρ is said to be orthogonal (resp. symplectic, of orthogonal type, of symplectic type) if Im(ρ) is. Recall that ρ ⊗ ρ = Sym2 (ρ) ⊗ Λ2 (ρ). We can relate the notions of being self-dual, orthogonal and symplectic as below. Proposition 2.1. The representation ρ is self-dual if and only if ρ ⊗ ρ contains the trivial character, and is essentially self-dual if and only if ρ ⊗ ρ contains a character. Further, ρ is of orthogonal (resp. symplectic) type if and only if Sym2 (ρ) (resp. Λ2 (ρ)) contains a character. If Sym2 (ρ) (resp., Λ2 (ρ)) contains the trivial character, then ρ is actually orthogonal (resp. symplectic). The endomorphism ring End(ρ) is isomorphic to the (self-dual) representation ρ ⊗ ρ̌. In particular, ρ ⊗ ρ̌ always contains the trivial representation. By Schur’s lemma, we have the following useful characterization. Proposition 2.2. The representation ρ ⊗ ρ̌ contains the trivial representation with multiplicity one if and only if ρ is irreducible. 2.2.2 Clifford’s Theorem Let V be a representation of a group G. We will say a subspace W ⊆ V is an eigenspace for G with character λ if there exists a character λ : G → C× such that g(w) = λ(g)w for all w ∈ W and all g ∈ G. Theorem 2.1. (Clifford) Let (ρ, V ) be a finite-dimensional irreducible representation of G and H E G. (i) ρH decomposes into irreducible representations of equal dimension, all of which are isomorphic if ρ is primitive. (ii) Suppose that V = ⊕Wi , where the Wi are eigenspaces for H with distinct characters. Then G/H permutes the Wi ’s. 11 Note that what is typically referred to as “Clifford’s theorem” is a more general and detailed statement about the decomposition of representations restricted to subgroups and does not usually include the specific result in (i) for when ρ is primitive. However this is a corollary of the usual Clifford’s theorem and just what we want. See, for example, [As] or [Gor]. 2.3 A Little Finite Group Theory Now we recall some standard definitions and results on finite groups. For proofs, see [As] or [Gor]. Let G be a finite group. Denote by Z(G) its center, Aut(G) its automorphism group, and by [G, G] its commutator. For S ⊆ G, let CG (S) denote the centralizer of S in G. For a prime number p, we write Sylp (G) for some Sylow p-subgroup of G. Definition 2.7. Suppose H E G and Q = G/H, i.e., we have a short exact sequence 1 −→ H −→ G −→ Q −→ 1. (2.6) Then we say that G is an extension of Q by H and write G = H · Q. Definition 2.8. Let G and H be finite groups with an injective homomorphism of centers φ : Z(G) ,→ Z(H). Let M = h(z, φ(z)) : z ∈ Z(G)i. The group (G × H)/M , denoted by G ◦ H or H ◦ G, is called a central product of G and H with identified centers. We remark that in general, given two groups G and H, there may be many central products G ◦ H. In the case that Z = Z(G) = Z(H) with Aut(Z) contained in both Aut(G) and Aut(H), the central product G◦H is uniquely defined. More generally, one can uniquely define a central product G1 ◦ G2 ◦ · · · ◦ Gr if Z(G1 ) = Z(Gi ) and Aut(Z(Gi )) ⊆ Aut(Gi ) for all i. In this case, taking central products is associative. We also need to define the following 2-groups of order 2m , each with a cyclic subgroup of index 2: —the dihedral group D2m = hx, y|x2 m−1 = y 2 = 1, yxy = x−1 i for m ≥ 2; —the semidihedral group SD2m = hx, y|x2 m−1 2m−1 —the quaternion group Q2m = hx, y|x = y 2 = 1, yxy = x2 m−2 −1 i for m ≥ 4; and 2m−2 = y 4 = 1, yxy = x−1 i/hx y 2 i for m ≥ 4. 12 2.3.1 The Fitting and Frattini Subgroups Recall that a subgroup H of G is called characteristic if H is invariant under Aut(G). Note that if H E G and K is a characteristic subgroup of H, then K E G. Definition 2.9. The Fitting subgroup of G is the maximal normal nilpotent subgroup of G, and is denoted by F (G). We remark that such a unique subgroup exists because the product of two normal nilpotent subgroups is again normal and nilpotent. The uniqueness evidently makes F (G) a characteristic subgroup of G. Proposition 2.3. If G is solvable, then CG (F (G)) ≤ F (G). Definition 2.10. The Frattini subgroup Φ(G) of G is the intersection of all maximal subgroups of G. Note that Φ(G) is also a characteristic subgroup of G. Proposition 2.4. Let P be a p-group. Then Φ(P ) is the smallest normal subgroup of P such that P/Φ(P ) is elementary abelian. 2.3.2 Extraspecial Groups Definition 2.11. Let P be a p-group. We say P is extraspecial if [P, P ] = Φ(P ) = Z(P ) and Z(P ) is cyclic. For an extraspecial p-group P , it follows that |Z(P )| = p and |P | = p2r+1 for some r. Let Qr8 D8s denote the central product of r copies of Q8 and s copies of D8 with all centers identified. There is a unique such group for given r, s. This is an extraspecial 2-group of order 22(r+s)+1 . As D8 ◦ D8 = Q8 ◦ Q8 , we see that Qr8 D8s = Qr+s−1 D8 or Qr+s depending 8 8 on whether s is odd or even. In fact any extraspecial 2-group of order 22r+1 is isomorphic r to either Qr−1 8 D8 or Q8 for some r. For any p-group P , the irreducible representations of P are monomial (i.e., induced from characters), and therefore have p-power dimension. For extraspecial p-groups, we have the following Proposition 2.5. Let P be an extraspecial group of order p2r+1 . Then all faithful irreducible complex representations of P are of degree pr . 13 Proposition 2.6. (Hall) A p-group P with no noncyclic characteristic abelian subgroups is a central product E ◦ R of two groups E and R, where E is trivial or extraspecial and either R is cyclic or R = D2m , SD2m or Q2m with m ≥ 4. 14 Chapter 3 Preliminaries on Galois and Automorphic Representations This chapter reviews theory of Galois and automorphic representations, and in particular the Artin and strong Artin (Langlands’s modularity) conjectures. 3.1 Galois Representations and Artin’s Conjecture By a Galois representation, we will always mean a continuous, complex, finite-dimensional representation of the absolute Galois group of a number field. Let F be a number field and GF = Gal(F /F ) be its absolute Galois group. Let ρ : GF → GLn (C) be a Galois representation, continuous for the profinite topology on GF . Such a representation factors through a finite quotient Gal(L/F ). The field L cut out by ρ is L = F ker ρ , i.e., the fixed field of ker ρ. Recall that ρ denotes the corresponding projective representation into PGLn (C). Then the field L0 cut out by ρ is L0 = F ker ρ . For a Galois representation ρ, we denote induction and restriction by the corresponding fields. Specifically if K ⊆ F ⊆ E are number fields, then ρE = ρGE denotes the restriction GK of ρ to GE and IndK F ρ = IndGF ρ denotes the induction of ρ to GK . When n = 1, then ρ is just a character χ. In fact, take any (possibly infinite order) character χ of GF . Then χ factors through the abelianization Gab F = GF /[GF , GF ]. Let CF = F × \A× F be the idèle class group of F . By class field theory, there is an isomorphism called the Artin map from CF to Gab F . There is a one-to-one correspondence between 15 characters χ of GF and idèle class characters (or Hecke characters) χ e of CF . Hence we may identify χ with χ e. We will sometimes simply say that χ or χ e is a character of F . In the case where χ is finite order and L is the field cut out by χ, then χ e factors through NL/F (CL )\CF , where NL/F is the norm map from L to F . If χ = χ e is non-trivial, we say it is an (idèle class) character of L/F . Now let us return to the general setting. Artin associated to ρ an L-function L(s, ρ) to study the primes of F . The Artin L-function generalizes the Dirichlet L-series and the Riemann zeta function. Specifically, for <(s) > 1 we can define L(s, ρ) by an Euler product, L(s, ρ) = Y Lv (s, ρ), (3.1) v<∞ where the product is over all finite places v of F and the local L-factors Lv (s, ρ) are given as follows. Let L be the field cut out by ρ and let w be a place of L above a finite place v. Denote the localization of F at v by Fv and its ring of integers by OFv . Let Frv ∈ Gal(L/F ) denote an element of the Frobenius conjugacy class at v, Iv the inertia group, and qv the size of the residue field |OLw /OFv |. Let V be the representation space of ρ. Denote by V Iv the space of inertial invariants, i.e., the subspace on which ρIv acts trivially. Then we may define the local L-factors by Lv (s, ρ) = 1
. det I − ρ(Frv )qv−s |V Iv (3.2) For almost all v, we have V Iv = V , in which case we say that ρ is unramified at v, or simply that v is unramified. At these places we just have Lv (s, ρ) = 1
. det I − ρ(Frv )qv−s (3.3) So defined, one can show that the Euler product (3.1) does indeed converge (in fact, absolutely and uniformly in compact sets) in <(s) > 1. Observe that if χ is a Dirichlet character, then L(s, χ) is in fact the Dirichlet L-series Q −s = −s −1 associated to χ. If 1 is the trivial representation on G , F n χ(n)n p (1 − χ(p)p ) P Q then L(s, 1) is the Dedekind zeta function a N(a)−s = p (1 − N(p)−s )−1 attached to F . P Q When F = Q, L(s, 1) is just the Riemann zeta function n n−s = p (1 − p−s )−1 . P Presently it will be convenient to view the local L-factors Lv (s, ρ) as L-functions L(s, ρv ) 16 of local representations. For any global or local field K, let WK denote the Weil group of K. Recall that this is a dense subgroup of the absolute Galois group of K. Now take K = Fv for some place v. Then WFv ⊆ Gal(Fv /Fv ) embeds in GF = Gal(F /F ) non-canonically, but the embeddings are conjugate. Fix one such embedding i : WFv ,→ GF . We can define the local representation i ρ ρv : WFv ,→ GF → GLn (C) by ρv = ρ ◦ i. As the Frobenius Frv is contained in WFv , it makes sense to define the L-function L(s, ρv ) for ρv by L(s, ρv ) = Lv (s, ρ). (3.4) Note that this local L-function depends only on ρ and v, not on the choice of the embedding i. Hence we will hereafter suppress the choice of i. One can also define L-factors at the infinite places. Let v be an infinite place of F . Either Fv = R or Fv = C. Define ΓR (s) = π −s/2 Γ(s/2) and ΓC (s) = 2(2π)−s Γ(s). If Fv = R let n+ = 21 (n + tr ρ(Frv )) and n− = 12 (n − tr ρ(Frv )). Then set   ΓR (s)n+ ΓR (s)n− , if Fv = R L(s, ρv ) =  Γ (s)n , if F = C. (3.5) v C One defines the completed Artin L-function to be L∗ (s, ρ) = Y L(s, ρv ), (3.6) v where the product runs over all places v of F . We write L∞ (s, ρ) for the product Q v|∞ L(s, ρv ) of local factors at the infinite places. Then L∗ (s, ρ) = L(s, ρ)L∞ (s, ρ). (3.7) Artin proved that L-functions are inductive, i.e., if ρ is induced from some representation 17 σ of a subgroup, then L(s, ρ) = L(s, σ). For any one-dimensional representation ρ, there exists by class field theory a Hecke character χ such that L(s, ρ) = L(s, χ). The Hecke L-function L(s, χ) has a functional equation relating L(s, χ) to L(1 − s, χ) of the form 1 γ(s, χ)L(s, χ) = W N 2 −s γ(1 − s, χ)L(1 − s, χ), (3.8) where |W | = 1, N ∈ Z+ and γ(s, χ) is a product of Γ-functions. More precisely, W is the “root number,” γ(s, χ) = L∞ (s, ρ) and N = Dn N(m) where D is the absolute value of the discriminant of F , N is the absolute norm and m is the module of definition of χ (i.e., χ is a primitive Größencharacter mod m). Hecke had proved that L(s, χ) is a meromorphic function on all of C, which is in fact analytic for s ∈ C − {1}. Moreover, L(s, χ) has a simple pole at s = 1 if χ is trivial and L(s, χ) is entire if χ is non-trivial. This Artin-Hecke correspondence implied that, for monomial representations ρ (i.e., those induced from one-dimensional representations), L(s, ρ) is meromorphic on all of C and entire unless ρ contains the trivial character. The functional equation (3.8) for L(s, χ) gives a functional equation for L(s, ρ) of the form 1 L∞ (s, ρ)L(s, ρ) = W (Dn N(f)) 2 −s L∞ (1 − s, ρ̌)L(1 − s, ρ̌), (3.9) i.e., 1 L∗ (s, ρ) = W (Dn N(f)) 2 −s L∗ (1 − s, ρ̌), (3.10) where W ∈ S 1 is the root number and f is the Artin conductor of ρ. Further, L-functions are additive: L(s, ρ ⊕ σ) = L(s, ρ)L(s, σ). So we also get the above results for direct sums of monomial representations. Brauer then showed that the character of any Galois representation ρ is a Z-linear comP bination of characters of monomial representations. Let us write ρ = mi IndFKi χi , where Q mi ∈ Z. Then the L-function L(s, ρ) = L(s, χi )mi admits a meromorphic continuation to the entire complex plane with a functional equation of the form (3.10) and has a pole at s = 1 whose order is the number of times the trivial representation occurs in ρ. Whence we arrive at 18 Conjecture 3.1. (Artin) Let ρ be a Galois representation. Then L(s, ρ) is analytic in C − {1}. In particular, L(s, ρ) is entire unless ρ contains the trivial character. We remark that, by the additivity of L-functions, to prove Artin’s conjecture for ρ, it suffices to prove it for all the irreducible constituents of ρ. By Proposition 2.2, we see that L(s, ρ ⊗ ρ̌) has a simple pole at s = 1 if and only if ρ is irreducible. Knowing Artin’s conjecture for all ρ would give us information about the primes of F . For instance, Artin’s conjecture has applications to bounding the error term in the Chebotarev density theorem (see Section 8 of Chapter 2 in [MM] for more information). Now suppose ρ : GF → GL2 (C). Let ρ : GF → PGL2 (C) be the composition of ρ with the canonical projection from GL2 (C) to PGL2 (C). Let G = Im(ρ). Then G is a finite subgroup of PGL2 (C). The finite subgroups of PGL2 (C) ≃ SO3 (C) have been classified by Klein, and there are five possibilties: (1) G is cyclic; (2) G is dihedral; (3) G is tetrahedral (isomorphic to A4 ); (4) G is octahedral (isomorphic to S4 ); or (5) G is icosahedral (isomorphic to A5 ). Accordingly, ρ is said to be cyclic, dihedral, tetrahedral, octahedral or icosahedral. In the cyclic case ρ is reducible, and in the dihedral case ρ is monomial. Thus the first two cases follow from the work of Artin, Brauer and Hecke. The next major step in this case was accomplished by Robert Langlands in [La2]. Langlands’s philosophy involves a vast generalization of abelian class field theory and Artin’s conjecture. We will not delve deeply into the so-called Langlands program, but only discuss Langlands’s modularity conjecture, which implies Artin’s conjecture. 3.2 Automorphic Representations and Langlands’s Modularity Conjecture Let F be a number field and AF be its ring of adèles, which is the restricted direct product Q0 v Fv over all places v of F with respect to {OFv }. Let G be an algebraic group over F . Q Then G(AF ) is the restricted direct product 0v G(Fv ) with respect to {G(OFv )}. One can 19 consider automorphic representations of G(AF ). We restrict ourselves to the case where G = GLn . In this case, the L-group of G is just L G = L G0 × W F , (3.11) where the connected component of the L-group is L G0 = GLn (C). (3.12) The Borel subgroup B is the subgroup of G comprising upper-triangular matrices. If v is non-archimedean then the maximal compact subgroup Kv of G(Fv ) = GLn (Fv ) is GLn (OFv ); otherwise Kv = O(n) or U(n) according to whether Fv is R or C. Let π be an automorphic representation of G(AF ), by which we mean an irreducible unitary representation of G(AF ) occurring in the space of automorphic forms on G(AF ). (Note that for n = 1, π is just a Hecke character.) Then π can be written as a restricted tensor product ⊗0v πv over all places v. Each πv is a local irreducible admissible representation of G(Fv ) such that, for almost all v, πv is unramified, i.e., the restriction of πv to Kv contains the trivial representation. If πv is unramified, then sometimes we will just say call v unramified (for π). At such a place, πv is induced from the Borel subgroup B(Fv ) of some tensor product µ1 ⊗ · · · ⊗ µn , where each µi is an unramified character of Fv× . If v is finite, let $ denote the uniformizer for Fv , i.e., a generator for the maximal prime ideal in Fv . Then one can associate to an unramified πv the semisimple conjugacy class in L G0 represented by A(πv ) = diag(µ1 ($), . . . , µn ($)). (3.13) The eigenvalues of A(πv ) are called the Satake parameters for πv . Here we may define the local L-function L(s, πv ) = 1 . det(I − A(πv )qv−s ) (3.14) Let S be a finite set of places of F containing the archimedean places and the places at which πv is ramified. Then we may formally define the incomplete automorphic L-function by an Euler product, LS (s, π) = Y v6∈S L(s, πv ) (3.15) 20 More generally, L(s, πv ) can be defined at any place and then we can formally define the complete automorphic L-function by L(s, π) = Y L(s, πv ). (3.16) v Denote the contragredient of π by π̌. Let D be the absolute value of the discriminant of F . For cuspidal π, let fπ be the conductor of π and W (π) the root number. Godemont and Jacquet showed the following Theorem 3.1. ([GoJ] or [Ja]) Let π be an automorphic representation of GLn (AF ). Then the Euler product L(s, π) converges in some right half-plane and L(s, π) meromorphically continues to the whole complex plane with a functional equation of the form L(s, π) = ε(s, π)L(1 − s, π̌). (3.17) The epsilon factor ε(s, π) is non-vanishing entire function, which is of the form 1 ε(s, π) = (Dn N(fπ )) 2 −s W (π) (3.18) when π is cuspidal. Moreover, if π is non-trivial and cuspidal, then L(s, π) is entire. Suppose π and π 0 are automorphic representations of GLn (AF ). Then by the work of many ([JS], [JPSS2], [Shah], [GS], [JS3], [CPS]) the Rankin-Selberg product L-function can be defined by L(s, π × π 0 ) = Y L(s, πv × πv0 ), (3.19) v where at unramified finite places v, the local factor is defined by L(s, πv × πv0 ) = 1 . det(I − (A(πv ) ⊗ A(πv0 ))qv−s ) (3.20) This Euler product for L(s, π × π 0 ) converges absolutely in the right half-plane <(s) > 1, meromorphically continues to the entire complex plane, is bounded in vertical strips, and has a functional equation of the form L(s, πv × πv0 ) = ε(s, π × π 0 )L(1 − s, π̌ × π̌ 0 ). (3.21) 21 The global ε-factor is given by an Euler product expansion ε(s, π × π 0 ) = Y ε(s, πv × πv0 , ψv ), (3.22) v where ψ is a non-trivial continuous additive character of AF which is trivial on F , and the local ε-factors are those defined in [JPSS2], [JS3] and [CPS]; the nonarchimedean local ε-factors are trivial whenever πv , πv0 and ψv are unramified. Using Rankin-Selberg products, Jacquet and Shalika [JS] showed that the Euler product for L(s, π) converges absolutely for <(s) > 1. Theorem 3.2. ([JS], [JPSS2], Appendix to [MW]) Let π (resp. π 0 ) be a cuspidal automorphic representation of GLm (AF ) (resp. GLn (AF )). Then L(s, π × π 0 ) is entire if and only if π 0 6≃ π̌. The only poles of L(s, π × π̌) are simple poles at s = 0 and s = 1. As a corollary to this theorem, we see that an isobaric representation π is cuspidal if and only if L(s, π × π̌) has a simple pole at s = 1. Theorem 3.3. ([La], [JS]) Let πi be an automorphic cuspidal representation of GLni (AF ) for 1 ≤ i ≤ r and set n = n1 +n2 +· · ·+nr . Then there exists an automorphic representation π = π1
π2
· · ·
πr of GLn (AF ), unique up to equivalence, such that L(s, πv ) = L(s, π1,v )L(s, π2,v ) · · · L(s, πr,v ), (3.23) for all places v. Consequently, L(s, π) = L(s, π1 )L(s, π2 ) · · · L(s, πr ). Such a sum π1
π2
· · ·
πr is called an isobaric sum. An automorphic representation is called isobaric if it can be written as a (finite) isobaric sum of cuspidal representations. The decomposition of an isobaric representation into cuspidal components is unique. Theorem 3.4. (Strong Multiplicity One [JS]) Let π and π 0 be isobaric automorphic representations of GLn (AF ). If πv ≃ πv0 at almost all places, then in fact π ≃ π 0 . Now we state the Langlands modularity conjecture, also called the strong Artin conjecture. Conjecture 3.2. Let ρ : GF → GLn (C) be a continuous Galois representation. Then there exists an isobaric automorphic representation π of GLn (AF ) such that L(s, ρv ) = L(s, πv ) for almost all places v of F . 22 If such a π exists, we will say that ρ is modular, or that ρ corresponds to π and write ρ ↔ π. Note that π is unique by Strong Multiplicity One. Whenever L(s, ρv ) = L(s, πv ), we say the local representations ρv and πv correspond, and we write ρv ↔ πv . The aforementioned corollary to Theorem 3.2 tells us that if ρ is irreducible, then π is cuspidal (cf. Proposition 2.2). The isobaric sum is the automorphic analogue of the direct sum of Galois representations in the sense that if ρi ↔ πi , 1 ≤ i ≤ r, then ρ1 ⊕ ρ2 ⊕ · · · ⊕ ρr ↔ π1
π2
· · ·
πr by Theorem 3.3. Thus it suffices to prove modularity for irreducible ρ. Now suppose that ρ is an irreducible representation and ρ ↔ π. Then π must be cuspidal. Also, any local factor L(s, ρv ) or L(s, πv ) is an entire function which is never zero. Hence L(s, ρ) differs from L(s, π) by a product of non-vanishing entire functions. Therefore L(s, ρ) is entire by Theorem 3.1. Thus the strong Artin conjecture for ρ indeed implies the Artin conjecture for ρ. Remarks. (1) If ρ ↔ π, then it is actually true that L(s, ρv ) = L(s, πv ) at all finite places v. In other words, L(s, ρ) = L(s, πf ) as Euler products (over F ), where πf denotes the restricted Q tensor product over all finite places ⊗0v<∞ πv and L(s, πf ) = v<∞ L(s, πv ). Moreover, the complete L-functions L∗ (s, ρ) and L(s, π) are equal as Euler products over Q. We give a proof of this, due to Ramakrishnan, in Appendix α. (2) The strong Artin conjecture holds for n = 1, and this is essentially just abelian class field theory as worked out by Artin. (3) Automorphic forms are generalizations of modular forms and Maass forms. For n = 2 and F = Q, the modularity of ρ would mean that ρ corresponds, by the work of Deligne and Serre [DS], to either a modular form of weight 1 (if ρ is odd) or a Maass form with Laplacian eigenvalue 14 (if ρ is even). (4) The strong Artin conjecture really is stronger (given what one knows today) than the Artin conjecture. For example, monomial representations are not known to be modular in general. However, for a fixed n = 2, 3, the Artin and strong Artin conjectures are globally equivalent. That is, knowing Artin’s conjecture for all n-dimensional representations is equivalent to knowing the strong Artin conjecture for all n-dimensional representations. Recently, Booker [Bo] showed for two-dimensional representations over Q that L(s, ρ) is 23 entire if and only if ρ is modular. Whether these two conjectures are equivalent in higher dimensons (either globally or for a specific representation) is not currently known. (5) The local Langlands correspondence for GLn (C) was proven recently by Harris and Taylor [HT] and subsequently by Henniart [He]. This means that, if ρ ↔ π, for each place v we can attach a local irreducible admissible representation πv to ρv . Thus ρ should correspond to π = ⊗0 πv . We know π is an irreducible, admissible representation but the difficulty lies is proving π is actually automorphic. The first (post-Artinian) cases of the strong Artin conjecture are stated in the celebrated Theorem 3.5. (Langlands [La2], Tunnell [Tu]) Let F be a number field and let ρ be a two-dimensional complex Galois representation of GF . If the image of ρ is solvable, then ρ is modular. The two main ingredients in the proof of this theorem, base change and the symmetric square lift, will be introduced in the next section. 3.3 Functoriality Results The general philosophy of Langlands is captured in the idea of functoriality, which we will Q not describe in general but only in specific cases. Consider H = i GLni , G = GLn as Q algebraic groups over F . Then L H 0 = i GLni (C) and L G0 = GLn (C). Let r be an algebraic homomorphism (an L-homomorphism) r : L H = L H 0 × WF → L G = L G0 × W F . (3.24) For an automorphic representation π of H unramified at a finite place v, there is an associated semisimple conjugacy class A(πv ). We say r is functorial if, for any automorphic representation π of H, we can transfer π to an automorphic representation π 0 of G such that A(πv0 ) = r(A(πv )) at all finite unramified places. Langlands’s functoriality conjecture states that all such r are functorial. We remark that by the local Langlands correspondence ([HT], [He]), if ρ ↔ π then a functorial map r will specify r(π) at all places by the correspondence r(ρ)v ↔ r(π)v . In our special cases, we will consider a homomorphism r : Q i GLni (C) → GLn (C) of the connected components of the L-groups. In general, the L-group is a semidirect product of 24 L G0 with W F , but here it is just a direct product. So r can be extended (in many ways) to a homomorphism of L-groups. In all of our examples below, excepting automorphic induction, we will extend r to L H by acting trivially on WF . The idea of functoriality and why it is important should be made clear through the following examples. 3.3.1 Base Change Key ideas in the proof of Theorem 3.5 relied on certain instances of base change for GL2 (cf. Introduction). Let ρ be a Galois representation of GF and let E/F be a finite extension. The idea is that if ρ ↔ π then there should be an automorphic representation πE of GLn (AE ), called the base change of π to E, such that ρE ↔ πE . In terms of functoriality, we mean that the restriction of scalars map from GLn (AF ) to GLn (AE ) ≃ GLn[E:F ] (AF ) should be functorial. Langlands [La2] proved normal cyclic base change for GL2 , which was subsequently generalized to GLn by Arthur and Clozel as described below. Let NE/F denote the norm map from E to F . Theorem 3.6. ([AC]) Let E/F be a normal cyclic extension of prime degree. For each isobaric representation π of GLn (AF ), there exists a unique isobaric automorphic representation of GLn (AE ) called the base change of π to E and denoted by πE such that (i) (descent) a cuspidal representation Π of GLn (AE ) is the base change πE of some π if and only if Π is Galois invariant (in particular, if ρE ↔ Π where ρ is some representation of Gal(L/F )); 0 if and only if (ii) if π 0 is also an isobaric representation of GLn (AF ) then πE = πE π 0 = π ⊗ δ for some idèle class character δ of NE/F (CE )\CF ≃ Gal(E/F ); (iii) (compatibility with reciprocity) if ρ is a representation of Gal(L/F ) such that ρ ↔ π and ρE is modular, then ρE ↔ πE ; and (iv) (compatibility with twisting) if χ is an idèle class character of F and the restriction χE = χ ◦ NE/F , then (π ⊗ χ)E = πE ⊗ χE . Note that, with the hypotheses of this theorem, if p = [E : F ], v is finite and unramified, and {a1 , a2 , . . . , an } are the Satake parameters for πv , then the Satake parameters for the 25 local base change πE,v are {ap1 , ap2 , . . . , apn }. The adjoint construction to base change is automorphic induction ([AC],[HH]), which corresponds to induction of Galois representations. Theorem 3.7. ([AC], [HH]) Let E/F be a normal cyclic extension of prime degree p and π an automorphic representation of GLn (AE ). Then there exists an automorphic representaF tion IF E π of GLnp (AF ), called the automorphic induction of π, such that L(s, π) = L(s, IE π) as Euler products over F . In particular, if ρ is a Galois representation of GF such that ρ ↔ π, then IndFE ρ ↔ IF E π. Jacquet, Piatetski-Shapiro and Shalika proved the following non-normal case of base change and automorphic induction. Theorem 3.8. ([JPSS]) Let E/F be a non-normal cubic extension of number fields. Let χ be an idèle class character of E and π an automorphic representation of GL2 (AF ). Then the automorphic induction IF E χ exists as an automorphic representation of GL3 (AF ) and the base change πE exists as an automorphic representation of GL2 (AE ). 3.3.2 Symmetric and Exterior Powers Let ρ : GF → GLn (C) be a continuous representation and let r : GLn → GLm be a symmetric or exterior power lifting. Then the local L-factors for the representation r(ρ) : GF → GLm (C) will just be L(s, r(ρ)v ) = det(I − r(ρ(Frv ))qv−s )−1 for finite unramified v. Analogously, at finite unramified v, one defines local automorphic L-factors L(s, πv ; r) = 1 det(I − r(A(πv ))qv−s ) (3.25) for automorphic representations π of GLn (AF ). In this setting, the functoriality conjecture says that one should be able to lift each π to an automorphic representation r(π) of GLm (AF ) such that L(s, r(π)v ) = L(s, πv ; r) at all unramified places v. In particular if ρ ↔ π, then r(ρ) ↔ r(π). Theorem 3.9. The following maps are functorial: (1) Sym2 : GL2 → GL3 ([GeJ]), (2) Sym3 : GL2 → GL4 ([KS]), (3) Sym4 : GL2 → GL5 ([Ki]), 26 (4) Λ2 : GL4 → GL6 ([Ki], [He2]). We remark that Kim [Ki] proved the exterior square Λ2 (π) equals an automorphic representation of GL6 (AF ) at all places, except possibly those above 2 and 3. This is actually enough for our purposes, since we only consider a correspondence at almost all places. However, we note that Henniart [He2] indicated how he can prove equality at the remaining places in question in a letter to Kim and Shahidi. 3.3.3 Tensor Products Fix positive integers m and n. Let π1 (resp. π2 ) be an automorphic representation of GLm (AF ) (resp. GLn (AF )). Let r be the tensor product map ⊗ : GLm × GLn → GLmn of F -groups. Functoriality of r means that there is an automorphic representation π1 ⊗ π2 of GLmn (AF ) such that L(s, (π1 ⊗ π2 )v ) = L(s, π1,v × π2,v ) for all v. In particular if ρ1 and ρ2 are Galois representations such that ρi ↔ πi , then ρ1 ⊗ ρ2 ↔ π1 ⊗ π2 . When m = 1, functoriality is known because if π is automorphic, then so is π ⊗ χ for any (unitary) character χ of F × \A× F. Theorem 3.10. The following maps are functorial: (1) ⊗ : GL2 × GL2 → GL4 ([Ra]) (2) ⊗ : GL2 × GL3 → GL6 ([KS]). 3.3.4 Applications to the Modularity Conjecture Modularity holds for the following kinds of Galois representations: (1) the direct sum of modular representations (2) the induction of a modular representation from a normal subgroup (3) the induction of a one-dimensional representation from a non-normal subgroup of index 3 (4) Symm (ρ) for two-dimensional modular ρ, m = 2, 3, 4 (5) Λ2 (ρ) for four-dimensional modular ρ (6) the tensor product of two modular representations of dimensions 2 and 2, or 2 and 3 27 (7) two-dimensional representations with solvable image (8) two-dimensional icosahedral representations over Q satisfying certain ramification criteria (9) the Asai lift of a modular two-dimensional representation (10) irreducible representations into GO4 (C) with solvable image (11) representations with nilpotent images. Note that (1)–(6) are direct applications of functoriality mentioned in the preceding sections. Case (7) is the aforementioned result of Langlands and Tunnell. Under case (8), various kinds of icosahedral representations have been shown to be modular. The works of Buzzard, Dickinson, Sheppard-Barron and Taylor [BDST]; Taylor [Ta]; and Goins [Goi] prove modularity for infinite families of icosahedral representations. For other approaches and examples, see also the works of Buhler [Bu]; Kiming and Wang [KW]; Jehanne and Müller [JM]; and Buzzard and Stein [BS]. Cases (9) and (10) were done by Ramakrishnan in [Ra2]. Case (11), which we state precisely in Proposition 9.1, was deduced from base change results by Arthur and Clozel [AC]. We emphasize that in all these instances, except perhaps (8), the proof relies heavily on some type of functoriality. Combining cases (8) and (4) gives examples of modular representations with non-solvable image in dimensions 3, 4 and 5. For primitive three-dimensional examples, see Section 8.1. Primitive four-dimensional examples are constructed using Sym3 in Section 10 of [KS]. A five-dimensional example induced from a non-normal quintic extension is constructed using Sym4 in [Ki2]. We remark that essentially the only cases where the Artin conjecture is known but the strong Artin conjecture is not are for the following types of Galois representations: (i) the induction of a modular representation from a non-normal subgroup (ii) the tensor product of two modular representations. Case (ii) follows from what is known about the Rankin-Selberg product of L-functions. In particular, if ρ1 , ρ2 and ρ3 are non-trivial, irreducible, modular representations of dimensions n1 , n2 and n3 , where n1 is 2 and n2 is 2 or 3, then the triple product ρ1 ⊗ ρ2 ⊗ ρ3 has an entire L-function by case (ii) and Theorem 3.10 above. 28 Chapter 4 The Method Let ρ : GF → GL4 (C) be a continuous representation with solvable image. We are concerned with the following question: How can we show ρ is modular? Here we will describe our basic approach to this problem. First recall that ρ has finite image by continuity, so L = F ker ρ is a finite Galois extension of F . Thus ρ factors through a representation of Gal(L/F ) with the same L-function. Hence sometimes we may view ρ as a faithful representation of Gal(L/F ) into GL4 (C). Let ρ : GF → PGL4 (C) be the composition of ρ with the standard projection from GL4 (C) to PGL4 (C). ρ / GL4 (C) II II II ρ II$  GQ I PGL4 (C) Let G denote the image of ρ and G denote the image of ρ. As in the GL2 case, we will classify ρ by its projective image G. To determine the possible cases for G, we will use the classification for certain subgroups of GL4 (C) and the group theory recalled in Chapter 2. Note that, conversely, for each finite solvable subgroup G of PGL4 (C), there is a representation ρ : GF → GL4 (C) with projective image G for some number field F . In fact, we may even take F = Q. This is because G is the image of some finite solvable subgroup G of GL4 (C). A (non-constructive) theorem of Shafarevich (Theorem 9.5.1 of [NSW]) says that if G is solvable, then there is a finite extension L/Q such that G = Gal(L/Q). Thus Gal(L/Q) (and hence GQ ) has a representation ρ into GL4 (C) with projective image G. We may restrict the problem of classifying subgroups of GL4 (C) to the corresponding problem for SL4 (C). The reason for this is as follows. Consider a subgroup G of GL4 (C) of 29 order n. Write ζm = e2πi/m . Let Z = hζ4n Ii be the cyclic group of scalar matrices of order 4n inside GL4 (C). Set G0 = ZG. Since we have only added scalar matrices to get G0 , then −k G0 = G. Given any g ∈ G, det g = ζnk for some k. Let z = ζ4n . Then  4 −k det zg = ζ4n det g = ζn−k ζnk = 1, (4.1) i.e., zg ∈ SLn (C). Now let H = G0 ∩ SL4 (C). Then any g ∈ G0 can be written as zh where z ∈ Z and h ∈ H. Therefore H = G0 = G, and G ⊆ ZH = G0 . Note the G is irreducible if and only if G0 is irreducible, which is if and only if H is irreducible. By the same logic, we also have that G is primitive if and only if H is primitive. So to determine the possibilities for the projective image of irreducible or primitive G, it suffices to assume that G is unimodular, i.e., that G is contained in SL4 (C). As an aside which we shall not have cause to use later (but we will morally use for the A5 case in Section 8.1), we observe that in fact G0 is both a central product of Z with G and of Z with H. For example, let C = Z(G) be the center of G. Then C ⊆ Z and G0 = Z ×C G. Let ρ be the standard representation of G. Then the standard representation of G0 is some twist ρ0 = χ ⊗ ρ, where χ is a character of Z. (More precisely, the tensored representation χ ⊗ ρ of Z × G factors through the representation ρ0 of G0 = Z ×C G.) Hence the representation ρ of G transfers to a representation ρ0H of H into SL4 (C) which is irreducible (primitive) if ρ is. 4.1 The Reducible Case Before moving on to the primary (irreducible) case, let us first remark on modularity of reducible four-dimensional representations of solvable type. Consider a reducible Galois representation ρ : GF → GL4 (C) where F is some number field. Say ρ is a direct sum of r irreducible representations τ1 , . . . , τr . Then ρ is modular if and only if each τi is. Any representation τi which is one-dimensional is modular. If we require that ρ have solvable image, then so does each τi . So in this case any two-dimensional τi is modular by the work of Langlands and Tunnell (Theorem 3.5). Thus the only interesting reducible case of solvable type is when ρ decomposes into a character χ and an irreducible three-dimensional representation τ . Then we are reduced to the modularity problem for τ . 30 Let τ : GF → GL3 (C) be an irreducible Galois representation of solvable type. Recall that if τ is monomial, then τ is modular by Theorem 3.8. Note that if τ is essentially self-dual, then the image of τ lies in GO3 (C). So Proposition 1.3 of [Ra2] tell us that τ must be monomial. Hence we may further assume that τ is a primitive three-dimensional representation (which, for solvable type, is necessarily non-essentially-self-dual). In Chapter 8, we examine the possible cases of such three-dimensional representations. There are no primitive three-dimensional τ of solvable type which are known to be modular (or to satisfy Artin’s conjecture). However, in certain cases, we are able to show that the associated eight-dimensional (self-dual) adjoint representation Ad(τ ) is modular. 4.2 The Irreducible Case We will now suppose that ρ is irreducible. Any scalar matrix in G lies in the center Z(G) of G. On the other hand, Schur’s lemma implies that the center of G is contained inside the scalar matrices. Therefore G = G/Z(G). As outlined in the introduction, our proofs of modularity will be similar to Langlands’s proof of the tetrahedral case. However whereas Langlands used the symmetric square lift Sym2 : GL2 → GL3 of Gelbart and Jacquet, we use the exterior square lift Λ2 : GL4 → GL6 of Kim and Henniart. Specifically, we can show the following. Proposition 4.1. Let ρ : Gal(L/F ) → GL4 (C) be an irreducible continuous representation with projective image G. Suppose that |G| = 2e p for some e ≥ 0 and an odd prime p, and that G has no elements of order 2p. Further suppose there is an automorphic representation π on GL4 (AF ) such that: (1) Λ2 (ρ) ↔ Λ2 (π); and (2) for some degree p subextension K/F of the field L0 cut out by ρ, the base change πK exists and ρK ↔ πK . Then ρ is modular. 31 L L0 2e G K p F Remark. As all the degree p subextensions K/F of L0 correspond to 2-Sylow subgroups of G, they are all conjugate. Proof. It suffices to show L(s, ρv ) = L(s, πv ) for almost all places v. Hence we will assume v is a finite place such that both ρv and πv are unramified. Then we can write down Frobenius eigenvalues {a, b, c, d} for ρv and Satake parameters {e, f, g, h} for πv . We want to show {a, b, c, d} = {e, f, g, h} . (4.2) As Λ2 (D) = 1 implies that D = ±I for a diagonal matrix D, condition (1) implies that {a, b, c, d} = ± {e, f, g, h} . (4.3) If these sets are equal, then we are done. So we may as well assume that {a, b, c, d} = − {e, f, g, h} . (4.4) Now we claim that the p-th powers of these sets are also equal. To see this, consider the e p-th power of the Frobenius, Frpv ∈ Gal(L/F ). For any x ∈ G, x2 p = 1 so xp is of 2-power order. Thus xp lies inside some 2-Sylow subgroup of G. By taking an appropriate conjugate of Frv , we may assume that Frpv ∈ Gal(L/K). Then ρ(Frpv ) = ρK (Frpv ). By Theorem 3.6(iii), we know ρK ↔ πK , which implies {ap , bp , cp , dp } = {ep , f p , g p , hp } . (4.5) 32 Without losing generality we may assume that a = ζp e and e = −b. Then a = −ζp b and     ρ(F rv ) ∼     −ζp b 0 0 0 0 0 0    b 0 0  .  0 c 0   0 0 d (4.6) Therefore     ρ(F rv ) ∼     −ζp 0 0 0 1 0 0 0 0 0 0 c b 0    0    0   (4.7) c a is an element of order divisible by 2p inside G. Contradiction! In the case that the extension K/F of degree p is normal, things are a bit simpler as we will now show. (In fact, this is the only case where we know condition (2) of the above proposition is satisfied.) Proposition 4.2. Let ρ : GF → GL4 (C) be an irreducible continuous representation with projective image G. Suppose that |G| = 2e p for some e ≥ 0 and an odd prime p, and that G has no elements of order 2p. Further suppose G has a normal 2-Sylow subgroup H. Let H be the pre-image of H in GF . Then ρ is modular if Λ2 (ρ) and ρH are modular. Proof. Let K/F be the subextension of L corresponding to H. Appealing to the previous proposition, we only need to show Λ2 (ρ) ↔ π and ρK ↔ πK for some automorphic representation π of GL4 (AF ). We know ρK corresponds to some automorphic representation Π of GL4 (AK ). Now we claim that Π is cuspidal, i.e., that ρK is irreducible. Suppose instead that ρK is a sum of irreducibles τ1 ⊕ · · · ⊕ τr with r > 1. Then K/F acts transitively on the τi ’s, so there must be r = p of them. On the other hand, Clifford’s theorem says they all have the same dimension m; so mp = 4. But p is assumed to be prime to 2. Contradiction. The extension K/F is normal and Π is cuspidal. By parts (i) and (ii) of Theorem 3.6, there are p automorphic representations π0 , π1 , . . . , πp−1 of GL4 (AF ) such that πi,K ≃ Π. 33 Furthermore, up to reordering, we may write πi ≃ π0 ⊗ δ i , (4.8) where δ is an idèle class character of K/F . Note that δ is a character of order p. Therefore the representations Λ2 (πi ) ≃ Λ2 (π0 ⊗ δ i ) ≃ Λ2 (π0 ) ⊗ δ 2i (4.9) are all distinct because they have distinct central characters ωΛ2 (πi ) = ωΛ2 (π0 ) δ 2i . (4.10) As δK is trivial, they all base change to Λ2 (π0 )K by Theorem 3.6(iv). Let β be the automorphic representation over F which is associated to Λ2 (ρ). Because Λ2 (ρ)K ↔ Λ2 (π0 )K , we also know that βK ≃ Λ2 (π0 )K . (4.11) However there are only p distinct automorphic representations which base change to Λ2 (π0 )K , so we must have β ≃ Λ2 (πi ) (4.12) for some i. This πi is our desired π. Another way we can demonstrate that ρ is modular is to replace the hypothesis that Λ2 (ρ) ↔ Λ2 (π) in Proposition 4.1 with the condition that ρM ↔ πM for a quadratic extension M/F . This kind of argument is given in Section 6.2. 34 Chapter 5 Solvable Primitive Subgroups of GL4(C) We are interested in (irreducible) primitive, solvable, finite subgroups G of SL4 (C). Such groups, including the non-solvable ones, were classified by Blichfeldt ([Bl]) in the early 1900’s. However, Blichfeldt’s list is presented in terms of generating matrices and geometrical invariants, which is not the most convenient form for us. We remark that Feit ([Fe]) presented a partial listing of such groups in a more modern form, but Feit’s list is also not quite sufficient for us. Thus we will rework, in large part, the classification of such solvable groups. Following a suggestion of D. Wales, we will study G by determining the possibilities for its Fitting subgroup. Then we follow methods of Blichfeldt to finish the classification we want. In particular we will show that G is either orthogonal or symplectic, and in the latter case there are precisely five possibilities for G. 5.1 The Fitting Subgroup Let G be a primitive, solvable, finite subgroup of SL4 (C). Let ρ be the standard representation of G on C4 . Proposition 5.1. Any normal abelian subgroup of G is cyclic of order 1, 2 or 4, and is contained in Z(G). In particular, |Z(G)| is 1, 2 or 4. Proof. Let A be a normal abelian subgroup of G. Then ρA is a sum of isomorphic characters χ ⊕ χ ⊕ χ ⊕ χ by Clifford’s theorem. Then A = ρ(A) is a subgroup of scalar matrices in G, 35 whence central. The only unimodular scalar 4 × 4 matrices are ±I and ±iI, so A must be cyclic of order 1, 2 or 4. Corollary 5.1. Any characteristic abelian subgroup of F (G) is cyclic of order dividing 4 and contained in Z(G). Proof. Let A be an abelian characteristic subgroup of F (G). Then A E G, whereby Proposition 5.1 applies. Proposition 5.2. The Fitting subgroup F (G) is a 2-group. Proof. Since F (G) is nilpotent, its Sylow subgroups Sylp (F (G)) are normal and F (G) is a direct product of them. Let p be any odd prime and P = Sylp (F (G)). Any automorphism of F (G) must leave P invariant, i.e., P is characteristic in F (G), whence normal in G. Then by Clifford’s theorem, ρP is a direct sum of irreducibles of the same degree. The degrees of the irreducible representations of P are powers of p. Thus ρP is a direct sum of characters. As ρ is faithful, P must be abelian. Hence P is trivial by Corollary 5.1. Proposition 5.3. The restricted representation ρF (G) is irreducible. Proof. Suppose otherwise, i.e., suppose that ρF (G) = τ1 ⊕ · · · ⊕ τr for r > 1. By Clifford’s theorem, either r = 2 and each τi is two dimensional, or r = 4 and each τi is one dimensional. Let Q = G/F (G). Because ρ is irreducible, Q acts transitively on {τi }. Let Q0 be the stabilizer in Q of τ1 and let G0 be the pre-image of Q0 in G. Note [G : G0 ] = [Q : Q0 ] = r. Also, by definition of Q0 , ρG0 decomposes as a direct sum of r representations, each of dimension 4r . Therefore, ρ is induced from G0 by one of the components of ρG0 , contradicting primitivity. Proposition 5.4. Either F (G) = E or F (G) = E ◦ C4 , where E is extraspecial of order 32. Proof. Write F (G) as a central product of two groups E and R as in Proposition 2.6. First we show that R must be cyclic. Suppose otherwise, i.e., that R = D2m , SD2m or Q2m for some m ≥ 4. Then R contains a cyclic subgroup X = hxi of order 2m−1 . By Proposition 2.4, every element of R/Φ(R) has order 2, so Φ(R) must contain the group hx2 i of order 2m−2 . On the other hand, Φ(R) is contained in X because X is maximal, and this containment must be strict as Φ(R) is the intersection of all maximal subgroups of R. Thus Φ(R) = hx2 i. 36 Clearly, Φ(R) ⊆ Φ(F (G)). Since Z(E) = Φ(E) = C2 , the quotient E/C2 is elementary abelian. Therefore (E × R)/(C2 × Φ(R)) is elementary abelian, from which (E ◦ R)/Φ(R) is also elementary abelian. Hence Proposition 2.4 yields that Φ(R) = Φ(F (G)). So Φ(R) is a characteristic abelian subgroup of F (G) of order 2m−2 . By Corollary 5.1, we must have that m = 4 and Φ(R) is a central subgroup of G (and R) of order 4. In particular |Z(R)| ≥ 4. But it is apparent from the definitions of D2m , SD2m and Q2m that |Z(R)| = 2. Contradiction! Wherefore R is cyclic. If R is cyclic of order 2m > 4, then it would be the only such subgroup of F (G), whence characteristic in F (G). Again Corollary 5.1 contradicts this. Thus R is cyclic of order 1, 2 or 4. Note that E ◦ C2 = E, so we may assume |R| = 1 or 4. It now suffices to show |E| = 32. By Proposition 5.3, we know that ρF (G) is irreducible. Thus ρE is also an irreducible four-dimensional representation. Proposition 2.5 then requires E to be of order 32. There are two extraspecial groups of order 32, which we denote by Q8 D8 and Q28 . The first is the central product of Q8 by D8 , and the second is the central product of Q8 by itself. Proposition 5.5. Q28 ◦ C4 = Q8 D8 ◦ C4 . Proof. Observe that for any finite group H and a cyclic group C, the central product H ◦ C is uniquely defined up to isomorphism. We first claim that Q8 ◦ C4 = D8 ◦ C4 . Write Q8 = hi, j : i4 = j 4 = (ij)4 = 1, iji−1 = j −1 i (5.1) D8 = hσ, τ : σ 4 = τ 2 = 1, τ στ = σ −1 i. (5.2) and Let C4 = hzi where z 4 = 1 and z commutes with Q8 and D8 . Define a map φ : Q8 ◦ C4 → D8 ◦ C4 (5.3) φ(z) = z, φ(i) = σ, φ(j) = zτ. (5.4) by first setting 37 Looking at the relations for Q8 and D8 , one sees that φ extends to a homomorphism of Q8 ◦ C4 into D8 ◦ C4 , which is actually an isomorphism. Then Q28 ◦ C4 = Q8 ◦ (Q8 ◦ C4 ) = Q8 ◦ (D8 ◦ C4 ) = Q8 D8 ◦ C4 . (5.5) (Note the central product is associative here because C4 is cyclic.) This implies there are only three cases for F (G): Q8 D8 , Q28 and Q28 ◦ C4 . We will write Q28 ◦ C4 simply as Q28 C4 . 5.2 The Structure of G Now we will be able to use this information about the Fitting subgroup to determine the structure of G. Specifically, G acts by conjugation on F (G), which gives us a homomorphism φ : G → Aut(F (G)) (5.6) with kernel Z(F (G)). The image of F (G) under this map φ will then be the inner automorphism group Inn(F (G)). Thus G/F (G) is isomorphic to a subgroup of the outer automorphism group Out(F (G)) = Aut(F (G))/Inn(F (G)). The outer automorphism groups of extraspecial groups are known. Specifically, Out(Q8 D8 ) ≃ A5 , Out(Q28 ) ≃ O+ 4 (F2 ) is the group of order 72 contained in S6 , and Out(Q28 C4 ) ≃ Sp4 (F2 ) × C2 is of order 1440. However we will only use the precise structure of Out(F (G)) at the end to simplify the proof of Proposition 5.7. Now, following the methods of Blichfeldt ([Bl]), we will embed G/F (G) inside S6 ≃ Sp4 (F2 ). As an extraspecial group of order 32 has a unique irreducible four-dimensional representation, we can write down a specific matrix representation for it, up to a change of basis. 38 Let  1 0    0 1 B1 =    0 0  0 0  0    1 B3 =    0  0 0 0    0 0   ,  −1 0   0 −1  1 0 0   0 0 0   ,  0 0 1   0 1 0  1    0 B2 =    0  0  0    0 B4 =    1  0 0 0 0    −1 0 0  ,  0 −1 0   0 0 1  0 1 0   0 0 1  ,  0 0 0   1 0 0 and B5 = iI be linear transformations of V = C4 with respect to a basis {v1 , v2 , v3 , v4 }. Then Q28 ≃ hB1 , B2 , B3 , B4 i (5.7) Q28 C4 ≃ hB1 , B2 , B3 , . . . , B5 i. (5.8) and Let X = hB1 , B2 , B3 , . . . , B5 i. Since C4 ⊆ Z(Q28 C4 ) and Q28 C4 is irreducible, there is only one way to extend Q28 to Q28 C4 , and that is by adjoining B5 . Thus we may assume Q28 C4 is represented by X. (Note that Q28 C4 actually has two inequivalent irreducible fourdimensional representations, but one may be obtained from the other by interchanging B5 with −B5 .) By Proposition 5.5, Q8 D8 is also contained in X = Q28 C4 . So in any case, we may assume that F (G) ⊆ X. 39 Note that one matrix representation of Q8 D8 is given by the group generated by  1 0 0 0       0 −1 0 0   , C1 = B1 B2 =     0 0 1 0    0 0 0 −1   0 i 0 0      i 0 0 0   ,  C3 = B3 B5 =    0 0 0 i    0 0 i 0  0 1    1 C2 = B1 B3 =    0  0  0    0 C4 = B4 B5 =    i  0 0 0 0 0 0 0 i 0 0    0  ,  0 −1   −1 0  i 0   0 i  .  0 0   0 0 0 Let {vij = vi ⊗ vj − vj ⊗ vi }1≤i<j≤4 be a basis for the six-dimensional vector space Λ2 (V ). Then Λ2 (ρ) provides an action of G on Λ2 (V ). In particular, the Bi ’s act as follows: · Λ2 (B1 ) v12 v13 v14 v23 v24 v34 v12 −v13 −v14 −v23 −v24 v34 Λ2 (B2 ) −v12 −v13 v14 v23 −v24 −v34 Λ2 (B3 ) −v12 v23 v14 Λ2 (B4 ) v24 v13 −v34 v34 −v13 −v23 −v14 −v24 v12 Λ2 (B5 ) −v12 −v13 −v14 −v23 −v24 −v34 Consider the basis {wi } for Λ2 (V ), where w1 = v12 + v34 , w2 = v12 − v34 , w3 = v13 − v24 , w4 = v13 + v24 , w5 = v14 + v23 , w6 = v14 − v23 . Let σ denote the composition of Λ2 (ρ) with the change of basis from {vi } to {wi }. Then 40 the Bi ’s act on the wj ’s as follows: · w1 w2 σ(B1 ) w1 w2 −w3 −w4 −w5 −w6 w3 w4 w5 w6 σ(B2 ) −w1 −w2 −w3 −w4 w5 σ(B3 ) −w1 −w2 −w3 w5 −w6 σ(B4 ) w4 w1 −w2 −w3 −w4 −w5 w6 w6 σ(B5 ) −w1 −w2 −w3 −w4 −w5 −w6 Thus the basis {wi } diagonalizes X, proving that σ(F (G)) ⊆ Diag(GL6 (C)). (5.9) Using the table above, we can also compute how the Ci ’s act on the wj ’s: · w1 w2 w3 σ(C1 ) −w1 −w2 w3 w4 w5 w6 w4 −w5 −w6 σ(C2 ) −w1 −w2 w3 −w4 −w5 w6 w2 w3 −w4 −w5 w6 σ(C3 ) w1 σ(C4 ) −w1 w2 w3 w4 w5 −w6 Proposition 5.6. G/F (G) embeds into S6 . Proof. Looking at the columns in the multiplication tables for the σ(Bi )’s and the σ(Ci )’s L above, we observe that σF (G) decomposes into six distinct characters. Thus Λ2 (V ) = Wi , where each Wi = hwi i is an eigenspace for F (G) with a distinct character. By Clifford’s theorem, G/F (G), acting by σ, permutes the six Wi . This permutation induces a map σ 0 : G → S6 . It remains to show that ker σ 0 = F (G). Let H = ker σ 0 . Clearly F (G) ⊆ H. Note that σ(H) ⊆ Diag(GL6 (C)), whence σ(H) is abelian. Since ker σ = {±I}, we have that H/ {±I} is abelian with {±I} ⊆ Z(H). Recall that a finite group is nilpotent if and only if each Sylow subgroup is normal. Hence if C is a central subgroup of a finite group N and N/C is nilpotent, then N is nilpotent. In particular, H is nilpotent. As H is a normal nilpotent subgroup of G, we must have H ⊆ F (G). Thus F (G) = H = ker σ 0 . 41 Now define  i 0 0 0   −i         0 1+i 0 i 0 0  1+i  S= √  , T =   2  1 2  0 0 1 0     0 0 0 0 1 0 0 i    1 1 0  ,  0 0 1   −i i 0 and J = hF (G), S, T i. We may extend σ to J so that σ(S) and σ(T ) act as the permutations (w1 w2 ) and (w2 w4 w6 w3 w5 ) respectively. Because hσ(S), σ(T )i = S6 , Proposition 5.6 implies that J is the largest finite subgroup of SL4 (C) which normalizes F (G). Thus G ≤ J. Let F20 be the group of order 20 contained in S5 , and let E = Q8 D8 , EC4 = Q28 C4 . (5.10) Note that F20 is the Frobenius group of order 20. (Recall that a group F is a Frobenius group if F acts transitively on a set S such that (i) each x ∈ F − {1} fixes at most one element of S and (ii) at least one x ∈ F − {1} fixes an element of S.) It contains D10 as a normal subgroup. Proposition 5.7. If G/F (G) has an element of order 5, then G is primitive and of symplectic type. In this case, there are precisely five possibilites for G up to isomorphism: (1) When F (G) = E, either G = E · C5 or G = E · D10 ; (2) When F (G) = EC4 , either G = EC4 · C5 , G = EC4 · D10 or G = EC4 · F20 . Note that this gives exactly three possibilities for the projective image G: E24 · C5 , E24 · D10 and E24 · F20 , where E24 denotes the elementary abelian group of order 24 . The fact that G ≤ J implies that these groups (both the projective ones and their pre-images) are in fact semidirect products. Proof. Suppose G/F (G) does contain an element of order 5. Then G contains an element α of order 5. We first show that G is primitive. As F (G) is a 2-group, it acts imprimitively on V . So we may suppose V = V1 ⊕ V2 ⊕ · · · ⊕ Vm , where m = 2 or 4, and F (G) acts transitively on the Vi ’s. If G is imprimitive, then G, and in particular α, must also permute the Vi ’s. Here α must act with order 1 or 5. Since there are less than five Vi ’s, the latter 42 is impossible. Thus α must fix each Vi , which contradicts α not commuting with F (G). Hence G is primitive. The only solvable subgroups of S6 containing an element of order 5 are C5 , D10 and F20 . Recall that G is of symplectic type if and only if Λ2 (G) contains an invariant line. So it suffices to show that σ 0 (G) acts intransitively on the lines Wi = hwi i. Clearly C5 cannot act transitively on 6 points. Since C5 E D10 of index 2, D10 cannot act transitively on 6 points either. As D10 E F20 of index 2, we see that F20 is also intransitive. To see that G is one of the five possible groups listed above, note that F (G) cannot be Q28 because Out(Q28 ) = G72 does not contain an element of order 5. Also Q8 D8 · F20 cannot occur because F20 is not contained in Out(Q8 D8 ) = A5 . Note that Q8 D8 · C5 and Q8 D8 · D10 occur as C5 and D10 are contained in Out(Q8 D8 ). The three possibilities for F (G) = Q28 C4 all occur as they are contained in J. For each of the extensions listed, there is only one possible isomorphism type as all subgroups of S6 which are isomorphic to C5 , D10 , or F20 are respectively conjugate. Proposition 5.8. If G/F (G) does not have an element of order 5, then G is of orthogonal type. Proof. See [Bl]. We remark that Blichfeldt’s classification (for solvable and non-solvable G) gives 30 types of projective images of primitive groups (up to conjugacy, not isomorphism) in GL4 (C): six simple, two with projective image S5 , one with projective image S6 , twelve of orthogonal type, and nine which are extensions of a subgroup H of S6 by E24 where H contains an element of order 5. 43 Chapter 6 The Primitive Cases Let F be a number field and ρ : GF → GL4 (C) be a continuous primitive Galois representation with image G and projective image G ⊆ PGL4 (C). Denote the projective representation by ρ. By Chapter 5, ρ is of either orthogonal or symplectic type. If ρ is of orthogonal type, then it is modular by [Ra2]. Let us suppose ρ is of symplectic type. Then by Proposition 5.7, either G = E24 · C5 , G = E24 · D10 or G = E24 · F20 . Let L (resp. L0 ) be the field cut out by ρ (resp. ρ). Let E be the subextension of L/F such that Gal(L0 /E) = E24 . Note that E/F is a normal extension. L L0 E24 G E C5 /D10 /F20 F 6.1 The Case E24 · C5 First we need to know the following about our group. Lemma 6.1. Every element g ∈ E24 · C5 has order 5, except for the elements in the normal subgroup E24 which have order 1 or 2. Proof. By Sylow’s theorem the number n5 of Sylow 5-groups in E24 · C5 is congruent to 1 mod 5 and also divides 16. Since E24 · C5 is not a direct product of E24 by C5 , a Sylow 44 5-subgroup is non-normal, i.e., n5 = 16. This accounts for 16 · 4 elements of order 5 in E24 · C5 . Hence the remaining 16 elements must comprise E24 . Now we can prove the main result of this section. Theorem 6.1. Suppose ρ : GF → GSp4 (C) ⊆ GL4 (C) has projective image G = E24 · C5 . Then ρ is modular. Proof. As above, L = F ker ρ , L0 = F ker ρ and now E ⊆ L0 is the normal quintic subextension of F such that Gal(L0 /E) = E24 . L L0 E24 G E C5 F First we will show that ρE is modular. As Gal(L/E) is a cyclic central extension of a 2-group, it is a direct product of a 2-group P2 with a cyclic group C of odd order. Therefore Gal(L/E) is nilpotent. A theorem of Arthur and Clozel (Proposition 9.1) states that all Galois representations with nilpotent image are modular. In particular ρE is modular. Now we claim that Λ2 (ρ) is also modular. Since ρ is of symplectic type, Λ2 (ρ) has an invariant line. Write Λ2 (ρ) = ν ⊕ r where ν is one dimensional and r is five dimensional. Note r is irreducible because it factors through E24 · C5 , which only has one- and fivedimensional irreducible representations. We claim r is induced from E. As P2 is a 2-group, each irreducible representation of P2 has dimension 2j for some j. Therefore the same is true for Gal(L/E) ≃ P2 × C. Hence in the decomposition of rE into its irreducible components, we must have a one-dimensional representation λ. In particular, r = IndFE λ. Since E is a normal subextension, r is modular by automorphic induction (Theorem 3.7), whence Λ2 (ρ) is also. Hence Proposition 4.2 implies the theorem. Remark. We know that examples of E25 ·C5 extensions of Q exist by Shafarevich’s Theorem (Theorem 9.5.1 of [NSW]) because E25 · C5 is solvable. Though Shafarevich’s proof is non- 45 constructive, we can illustrate how to construct such an extension in our case. Let ζ11 be a i + ζ −i . Let E be the cyclic Galois extension primitive 11-th root of unity and set αi = ζ11 11 Q(α1 ) of Q of degree 5. It is known how to construct Q8 and D8 extensions of a number √ √ √ field ([JLY]). Let K = E( 1 + A + B + AB) and M = E( α1 + iα2 , α2 + α4 + 4), where A = (3+α5 )−1/2 and B = (1+α12 +α12 α32 )−1/2 . Then Gal(K/E) ≃ Q8 and Gal(M/E) ≃ D8 . The compositum KM has three normal (over E) subextensions of index 2. Let L/E be the one corresponding to the central product of Q8 with D8 . Then L/Q is Galois with Galois group E25 · C5 . 6.2 The Cases E24 · D10 and E24 · F20 We keep the notation from the start of the chapter. First suppose Gal(L0 /F ) = E24 · D10 . Then there exists a (normal) quadratic extension M of F contained in L0 such that Gal(L0 /M ) = E24 ·C5 and there are five non-normal quintic subextensions K1 , K2 , K3 , . . . , K5 of L0 /F . L L0 E24 EB BB BBC2 BB B C5 Ki G D10 MB 5 BB BB B C2 BB F Definition 6.1. We will say that condition (NNQ) is satisfied for L/F if: (NNQ) For any automorphic representation π of GL4 (AF ) such that πM is associated to a representation of Gal(L/M ), the base change πKi exists as an automorphic representation for some i = 1, 2, 3, . . . , 5. 46 Theorem 6.2. Suppose ρ : GF → GSp4 (C) ⊆ GL4 (C) has projective image G = E24 · D10 . Further suppose that condition (NNQ) holds for L/F . Then ρ is modular. Proof. By Theorem 6.1, there are two automorphic representations π1 , π2 of GL4 (AF ) such that ρM ↔ πi,M . Let δ = δM/F be the idèle class character of M/F . Then π1 ≃ π2 ⊗ δ. (6.1) Let K ⊆ L be a non-normal quintic extension of F such that the base change πK exists by (NNQ). We claim that ρK ↔ πi,K (6.2) for precisely one of the πi ’s. First note that Gal(L0 /K) is a 2-group. Thus Gal(L/K), being a cyclic central extension of Gal(L0 /K), is the direct product of a 2-group and a cyclic group of odd order. In particular, Gal(L/K) is nilpotent. Hence ρK is modular. Let us say Π is the automorphic representation of GL4 (AK ) such that ρK ↔ Π. (6.3) We want to know that Π is cuspidal, i.e., that ρK is irreducible. Let H be the Fitting subgroup F (Gal(L/F )). Because Gal(L/K) is a maximal nilpotent subgroup, we have H = F (Gal(L/F )) ⊆ Gal(L/K). (6.4) By Proposition 5.3, ρH = (ρK )H is irreducible, wherefore ρK is also. (Though all of our groups were unimodular in Chapter 5, the proof of Proposition 5.3 still works without the unimodularity assumption.) Similarly, ρE is irreducible because Gal(L/E) = F (Gal(L/M )) is. Thus Π and ΠE are cuspidal. By Theorem 3.6(i),(ii), there are precisely two automorphic representations of GL4 (AK ) whose base change to E is ΠE . They are Π and Π ⊗ δE/K , where δE/K is the idèle class character of E/K. As π1,E , π2,E and ΠE all correspond to ρE , they must be equal by Strong Multiplicity One. So it must be that   Π, Π ⊗ δE/K = π1,K , π2,K ≃ π1,K ⊗ δE/K . (6.5) 47 Thus Π ≃ πi,K for some i. Denote this πi simply by π, so then ρM ↔ π M , ρ K ↔ π K . (6.6) We want to show that ρv ↔ πv for almost all places v. Let v be a finite place for which both ρv and πv are unramified. Let {a1 , a2 , . . . , a4 } be the eigenvalues of ρ(Frv ) and {b1 , b2 , . . . , b4 } those of A(πv ). We want to show these sets are equal. As any element of D10 squared lies inside C5 , we have that Fr2v ∈ Gal(L/M ). Thus  ρ(Fr2v ) = ρM (Fr2v ) ∼ a21 , a22 , . . . , a24 . (6.7) Since ρM,v ↔ πM,v , we have that  2  a1 , . . . , a24 = b21 , . . . , b24 . (6.8) a2i = b2i (6.9) We may further assume that for each i = 1, 2, . . . , 4. On the other hand, Fr5v ∈ Gal(L/K 0 ) for some conjugate K 0 ∈ {Ki } of K. We may choose an appropriate element from the Frobenius class to assume that K 0 is in fact K. Then  ρ(Fr5v ) = ρK (Fr5v ) ∼ a51 , a52 , . . . , a54 . (6.10)  5  a1 , . . . , a54 = b51 , . . . , b54 . (6.11) Hence If {ai } = {bi } then we are done. So we may assume that they are not. Without loss of generality, suppose that b1 = −a1 , br = ζ5 as , (6.12) for some r, s and a primitive fifth root of unity ζ5 . We want to deduce that ρ(Frv ) has order divisible by 10, which would contradict the fact that G has no elements of order 10. Note that we may assume ai 6= −ζaj (6.13) 48 for any primitive fifth root of unity ζ. For otherwise we would have that  ak al ρ(Frv ) ∼ diag 1, −ζ, , ai ai  (6.14) is an element of order 10 in G, where k and l are appropriate indices. First suppose s = 1. Then we may assume r = 2 so that b2 = ζ5 a1 . If b2 = −a2 , then a2 = −ζ5 a1 , opposing assumption (6.13). Hence b2 = a2 = ζ5 a1 . Note b1 = −a1 = −ζ5−1 a2 so b51 6= a52 . Interchanging a3 and a4 if necessary, we then have a53 = b51 = −a51 . By assumption (6.13) again, this means that a3 = −a1 . Thus   a4 ρ(Frv ) ∼ diag (a1 , ζ5 a1 , −a1 , a4 ) ∼ diag 1, ζ5 , −1, a1 (6.15) is an element of order divisible by 10. Contradiction! Now we may assume s = 2 and either r = 1 or r = 3. If r = 1, then a1 = −b1 = −ζ5 a2 contradicting (6.13). Thus r = 3 and b3 = ζ5 a2 . By (6.13), we cannot have b3 = −a3 , so b3 = a3 . Then (6.11) reads  5 5 5 5  a1 , a2 , a2 , a4 = −a51 , b52 , a52 , b54 . (6.16) Either −a51 = a52 or −a51 = a54 . Consider first −a51 = a52 . Then a1 = −a2 and   a4 ρ(Frv ) ∼ diag (−a2 , a2 ζ5 , a2 , a4 ) ∼ diag −1, 1, ζ5 , a2 (6.17) has order divisible by 10. Thus we can consider the (final!) case −a51 = a54 , i.e., a1 = −a4 . Here we also have that  ρ(Frv ) ∼ diag (a1 , a2 , ζ5 a2 , −a1 ) ∼ diag a1 a1 , 1, ζ5 , − a2 a2  (6.18) is an element of G of order divisible by 10. In any case, we get a contradiction, so we are done! Now consider Gal(L0 /F ) = E24 · F20 . Then there exists a (normal) quadratic extension M of F inside L0 such that Gal(L0 /M ) = E24 · D10 and there are five non-normal quintic subextensions K1 , K2 , K3 , . . . , K5 of L0 /F . (See the diagram below.) The definition of 49 condition (NNQ) for L/F also makes sense in this case, and by exactly the same method of the proof of Theorem 6.2 we can also prove the following. Theorem 6.3. Suppose ρ : GF → GSp4 (C) ⊆ GL4 (C) has projective image G = E24 · F20 . Further suppose that condition (NNQ) holds for L/F . Then ρ is modular. L L0 E24 EB BB BBC4 BB B D10 Ki G F20 MB 5 BB BB B C2 BB F 6.3 Transfer to GSp4 Consider any irreducible four-dimensional Galois representation ρ : GF → GSp4 (C). The Langlands philosophy predicts that ρ should correspond not only to an automorphic representation of GL4 (AF ), but also to an automorphic representation of the group GSp4 (AF ) (since L GSp04 = GSp4 (C)). Suppose that ρ is modular, i.e., that ρ corresponds to some cuspidal representation π of GL4 (AF ). The fact that Im(ρ) ⊆ GSp4 (C) implies that L(s, Λ2 (ρ) ⊗ ν −1 ) has a simple pole at s = 1 for a suitable one-dimensional representation ν of G (the “polarization”). This implies that the corresponding automorphic L-function L(s, π; Λ2 ⊗ ν −1 ) has a pole at s = 1. An unpublished theorem of Jacquet, Piatetski-Shapiro and Shalika says that, because of this pole, π transfers to a generic irreducible cuspidal automorphic representation Π of GSp4 (AF ) with central character ν such that, LS (s, Π) = LS (s, π), 50 for any finite set of primes S outside of which π is unramified. Here the L-function on Q the left is the degree four L-function of Π studied in [PS]; and if L(s) = v Lv (s) is an Q Euler product, then LS (s) denotes the incomplete L-function v6∈S Lv (s). Thus ρ in fact corresponds to the cuspidal representation Π of GSp4 (AF ), i.e., ρ is modular of symplectic type as predicted by Langlands. In particular, the representations in Theorem 6.1 should actually correspond to automorphic representations on GSp4 (AF ). However we are not stressing this here because this theorem of Jacquet, Piatetski-Shapiro and Shalika remains unpublished. We hope to work this out in the future, which should not be too difficult using some recent results. The key point is that GL4 maps into the connected component of GO6 and π gives rise to a cuspidal automorphic representation π 0 of GO6 (AF )0 . The desired Π is obtained by the theta correspondence. The obstruction to this transfer is the residue of the pole of L(s, π; Λ2 ⊗ ν −1 ) at s = 1 (see [JS2]). Finally the ongoing work of J. Arthur will give another proof, using the trace formula, of the existence of Π and other members of its packet (see [Ar] for his program). 51 Chapter 7 Monomial Representations Suppose now ρ : GF → GL4 (C) is an irreducible, continuous, imprimitive representation with solvable image, where as before F denotes a number field. Either ρ is induced from a character or a two-dimensional representation (or both). If ρ is induced from a twodimensional representation σ, then the fact that σ is modular (Theorem 3.5), combined with automorphic induction (Theorem 3.7), yields that ρ is also modular. Thus we will assume that ρ is not induced from a two-dimensional representation. Hence ρ is monomial, i.e., there is a quartic extension K of F and a character χ of K such that ρ = IndFK χ. (7.1) If K/F is normal, then ρ is modular by Theorem 3.7. So assume K/F is non-normal. If E is an intermediate subfield of K/F , then ρ is induced by the two-dimensional representation σ = IndE K χ of GE , going against our assumption above. Hence K/F has no intermediate subfields. In this case, Artin’s conjecture is known, but the strong Artin conjecture is not. 7.1 The Structure of Im(ρ) We keep the assumptions above, i.e., that (i) ρ = IndFK χ for some character χ, and (ii) K/F is a non-normal quartic extension with no intermediate subfields. e the Galois closure of K/F , and L the Galois closure Let K(χ) be the fixed field of ker χ, K 52 of K(χ)/F . L e K(χ)K EE EE EE EE E u uu uu u u uu K(χ) JJ JJ JJ JJ JJ K ww ww w w ww ww e K 4 F Then ρ factors through GL . In fact, ρ : Gal(L/F ) → GL4 (C) is faithful, for L = F ker ρ (7.2) is a normal (Galois) extension of F . Proposition 7.1. Suppose ρ : GF → GL4 (C) is monomial, induced from quartic extension e ) is isomorphic K/F , and ρ is not induced from any quadratic extensions. Then Gal(K/F e to either A4 or S4 , where K/F is the Galois closure of K/F . e ) is a subgroup of S4 with order divisible by 4 but not equal Proof. We know that Gal(K/F e )| must be 8, 12 or 24. If |Gal(K/F e )| = 8, then Gal(K/K) e to 4. Thus |Gal(K/F is contained e ) of index 2. However, M would correspond to an in a maximal subgroup M of Gal(K/F e )| = 12 or 24, in which intermediate subfield of K/F , giving a contradiction. So |Gal(K/F e ) ≃ A4 or S4 . case Gal(K/F Note that as ρKe contains a character, it must be the direct sum of four characters by e is an abelian group A of rank at most 4. Clifford’s theorem. Therefore Gal(L/K) Let G = Gal(L/F ). Then G is of the form A · A4 or A · S4 . In fact, we can say a bit more. Lemma 7.1. With ρ, G and A as above, the Fitting subgroup F (G) = P2 × B where P2 is a 2-group and B is an abelian group of odd order. Proof. The Fitting subgroup F (G) is a product Q Sylp (F (G)) of Sylow p-subgroups. Let p be an odd prime and P = Sylp (F (G)). We want to show that P is abelian. As P 53 is characteristic in F (G), it must be normal in G. Therefore, ρP decomposes a sum of representations whose dimensions are all equal by Clifford’s theorem. On the other hand, any irreducible representation of P has p-power dimension. Thus the only possibility is that ρP decomposes as a sum of characters. Since ρ is faithful, this implies that P is abelian. Lemma 7.2. Let ρ : GF → GL4 (C) be an irreducible representation. If Λ2 (ρ) contains two one-dimensionals, then ρ is induced by a two-dimensional representation τ along a quadratic extension. Proof. First note that Λ4 (ρ) is the determinant det(ρ). Let V be the representation space for ρ. We have a natural pairing, Λ2 (ρ) × Λ2 (ρ) −→ Λ4 (ρ) = det(ρ), (7.3) hΛ2 (ρ)(g)v, Λ2 (ρ)(g)wi = det(ρ)(g)hv, wi ∀g ∈ GF , ∀v, w ∈ Λ2 (V ), (7.4) i.e., where h , i denotes an orthogonal form on Λ2 (V ) ≃ C6 . Thus Λ2 (ρ) an essentially selfdual representation for any four-dimensional ρ, whether or not ρ is itself essentially selfdual. Moreover, if ν is any polarization, i.e., a one-dimensional summand of Λ2 (ρ), then ν 2 = det(ρ). This can be seen by restricting (7.4) to the line hv0 i ⊆ Λ2 (V ) which is ν-invariant, i.e., ν(g)2 hv0 , v0 i = hν(g)v0 , ν(g)v0 i = hΛ2 (ρ)(g)v0 , Λ2 (ρ)(g)v0 i = det(ρ)(g)hv0 , v0 i ∀g ∈ GF . (7.5) Suppose ν 0 is another character occurring in Λ2 (ρ) and let µ = ν −1 ν 0 . Then µ2 = ν −2 (ν 0 )2 = det(ρ)−1 det(ρ) = 1. (7.6) We claim that µ 6= 1, i.e., ν 6≃ ν 0 . First observe that, because ν is contained in Λ2 (ρ), ρ is of symplectic type for some symplectic form h , iν such that hρ(g)v, ρ(g)wiν = ν(g)hv, wiν ∀g ∈ GF , ∀v, w ∈ V, (7.7) 54 i.e., ρ̌ ≃ ρ ⊗ ν −1 . Therefore End(ρ) = ρ ⊗ ρ̌ ≃ ρ ⊗ ρ ⊗ ν −1 contains Λ2 (ρ) ⊗ ν −1 . If ν 0 ≃ ν, then Λ2 (ρ) ⊗ ν −1 , and hence ρ ⊗ ρ̌, contains the trivial representation with multiplicity at least two, contradicting the irreducibility of ρ (Proposition 2.2). Hence µ 6= 1. Now let M be the quadratic extension of F which is cut out by µ. Then Λ2 (ρM ) contains νM with multiplicity 2. The argument in the previous paragraph applies here to yield that ρM is reducible. Hence ρ is induced from a two-dimensional representation defined over M. Proposition 7.2. Suppose that ρ : GF → GL4 (C) a Galois representation of solvable type which is imprimitive and essentially self-dual. Then ρ is modular. Proof. An essentially self-dual, reducible, four-dimensional representation of solvable type is always modular (Section 4.1). Therefore, assume that ρ is irreducible. Either ρ is of orthogonal or symplectic type. If ρ is of orthogonal type, then it is modular by the recent work of Ramakrishnan [Ra2]. So let us further assume that ρ is of symplectic type. Thus Λ2 (ρ) decomposes into Λ2 (ρ) = ν ⊕ τ, (7.8) where ν is the polarization character and τ is a five-dimensional representation. By earlier remarks, we may assume that ρ = IndFK χ for some non-normal quartic extension K/F and a character χ of GK . We claim that ρ is induced from a two-dimensional e ) ≃ A4 or Gal(K/F e ) ≃ S4 . By Lemma 7.2, representation. Suppose it is not. Then Gal(K/F Λ2 (ρ) cannot contain two characters, i.e., τ cannot contain a character. Because the restrice ) tion ρKe decomposes as a sum of characters, so does τKe . If τ were irreducible, Gal(K/F would act transitively on the five characters contained in τK , which is impossible. Hence τ is reducible and must decompose as τ = α ⊕ β, (7.9) where α is an irreducible two-dimensional representation and β is an irreducible threedimensional representation. e ) ≃ A4 . Let M/F be the subexNow we essentially reduce to the case where Gal(K/F e e ). So if Gal(K/F e ) ≃ A4 tension of K/F corresponding to the subgroup A4 of Gal(K/F e ) ≃ S4 then M/F is quadratic. In either case βM must be then M = F and if Gal(K/F 55 irreducible, using the fact that Gal(M/F ) acts transitively on the irreducible components e of βM . Let N be the subextension of the A4 -extension K/M corresponding to a cyclic subgroup of order 3. e K C3 A4 N 4 A4 /S4 M 1/C2 F Thus N/M is a (non-normal) quartic extension and we can also show that βN is irreducible. To see this, first note that βKe is a sum of characters because ρKe is. The Galois e ) on the characters in β e implies that either βN is irreducible or it is a action of Gal(K/N K e ) stabilizes each comsum of characters µ1 ⊕ µ2 ⊕ µ3 . Suppose the latter. Then Gal(K/N o n e ponent µi,Ke of βKe . But Gal(K/M ) acts transitively on µi,Ke because βM is irreducible. e In fact the action of Gal(K/M ) must factor through C3 since S3 is not a factor of A4 . Now e ) is a maximal subgroup of Gal(K/M e as Gal(K/N ), it must be the stabilizer of each µi,Ke . Hence we have a short exact sequence C3 A4 z }| { z }| { e )−→Gal(K/M e )−→ C3 −→ 1, 1 −→Gal(K/N (7.10) which is just wrong in so many ways! (well, at least two). Thus βN is indeed irreducible. Note that as ρKe is a sum of characters, we can write ρN = λ ⊕ σ, where σ is a (possibly reducible) three-dimensional representation. Then σKe is also a sum of characters. As the e Galois group of K/N acts on the summands of σKe , the representation σ must either be irreducible or itself a sum of characters. If σ is a sum of characters, then so are ρN and Λ2 (ρN ), contradicting the irreducibility of βN . Thus σ is irreducible. Now on one hand we have Λ2 (ρN ) = Λ2 (χ ⊕ σ) = Λ2 (σ) ⊕ χσ. (7.11) 56 On the other hand, Λ2 (ρN ) = Λ2 (ρ)N = νN ⊕ αN ⊕ βN . (7.12) Since χσ is irreducible, dimensions require that χσ = βN and therefore Λ2 (σ) = νN ⊕ αN (7.13) is reducible. But the fact that we have a natural pairing Λ2 (σ) × σ ≃ Λ2 (σ) × Λ(σ) −→ Λ3 (σ) ≃ det(σ) (7.14) implies that Λ2 (σ) is a twist of σ, whence irreducible, contradicting (7.13). Thus ρ is induced from two-dimensional representation, and therefore modular. 7.2 Hypertetrahedral Representations In this section, we consider the case where G is an extension of A4 by a group of order 4. Such groups can be easily computed in [GAP] (or even rather easily, but more tediously, by hand). There are six possibilities for G : C4 × A4 , V4 × A4 , SL2 (F3 ) × C2 , SL2 (F3 ) o C2 , V4 o A4 and V4 · A4 . Here V4 is the Klein 4-group, and V4 · A4 is the unique group of order 48 containing both V4 and A4 as subgroups which is not a semidirect product of the two. In all of these cases, ρ is modular. However, the first four cases are the less interesting ones, as we will see anon. First we need the Proposition 7.3. Let ρ be an irreducible four-dimensional complex representation of GF such that G is one of the six possible extensions of A4 by C4 or V4 . Then ρ is monomial. e More precisely, let L be the fixed field of ker(ρ), and let K/F be a subextension of L/F corresponding to the quotient group A4 . Let K/F be a non-normal quartic subextension of e . Then ρ is induced from a character of GK . K/F e Proof. Let E be the intermediate field of K/F corresponding to the subgroup V4 . Then E/F is a normal extension of degree 3. Let L0 be the field cut out by ρ. (See the diagram below.) We claim that ρ is induced from K, i.e., that ρK contains a character. Assume otherwise. e = C4 or V4 , any irreducible representation of Gal(L/K) e has dimension 1 Since Gal(L0 /K) 57 or 2. Thus ρK cannot be irreducible since the restriction ρKe to a normal cubic extension is not. So we may assume that ρK is a sum of two irreducible two-dimensional representations. Restricting, we see that ρKe is also sum of two irreducible two-dimensional representations, e ) = A4 acts transitively on {σ, τ }. So the say ρKe = σ ⊕ τ . As ρ is irreducible, Gal(K/F stabilizer of σ in A4 is a subgroup of index 2. But A4 has no subgroups of index 2, a contradiction. L cyclic L0 P2 ×C C4 /V4 Q2 } } } } }} KB BB BB B 4 BB C3 }} e @ K @ @@ V4 @@ @@ A4 F }} }} } } }} C3 E Proposition 7.4. Suppose ρ : GF → GL4 (C) is a Galois representation whose projective image G is an extension of A4 by a group of order 4. If ρ is reducible, then ρ is modular. If G ≃ C4 × A4 , V4 × A4 , SL2 (F3 ) × C2 , or SL2 (F3 ) o C2 , then ρ is reducible. Proof. Note that ρ is of solvable type. Suppose that ρ is reducible. Then ρ is modular except possibly when ρ decomposes as a character χ plus a three-dimensional τ (see Section 4.1). However A4 has a normal subgroup of index 3, which means that G does also. Denote the corresponding cubic extension of F by E. Let L be the field cut out by ρ and L0 be the field cut out by ρ. Then Gal(L0 /E) is a 2-group Q2 , so the cyclic central extension Gal(L/E) is the direct product of a 2-group P2 with a cyclic group C of odd order. (See diagram above.) In particular, any irreducible representation of Gal(L/E) has dimension a power of 2. Thus ρE must contain a linear character χ. Hence, ρ is induced by χ along the normal cubic extension E/F . Theorem 3.7 then implies that ρ is modular. e ) = A4 acts transitively Now suppose that ρ is irreducible. The Galois group Gal(K/F e ) cannot fix on the four distinct characters occuring in ρKe . This implies that Gal(K/F e pointwise. However, as may be checked in [GAP] for example, each of the four Gal(L0 /K) 58 groups C4 × A4 , V4 × A4 , SL2 (F3 ) × C2 and SL2 (F3 ) o C2 has a unique normal subgroup of order four whose quotient is A4 . But in all of these cases, this normal subgroup is the e pointwise. This shows that center of G, i.e., every element of Gal(L0 /F ) fixes Gal(L0 /K) G = V4 o A4 or V4 · A4 . If ρ is irreducible and G is an extension of A4 by V4 , then we will say that ρ is hypertetrahedral. Hypertetrahedral representations occur for both of the remaining cases G ≃ V4 o A4 and G ≃ V4 · A4 . We remark however that such a projective image does not guarantee irreducibility. For example, V4 · A4 has a faithful irreducible three-dimensional representation τ . Since V4 · A4 is centerless, 1 ⊕ τ is a reducible four-dimensional representation with projective image V4 · A4 . Now we come to the main result of this chapter: Theorem 7.1. Let F be a number field and ρ a hypertetrahedral representation of GF . Then ρ is modular. There are infinitely many such representations with projective image V4 · A4 which are not essentially self-dual. Remarks. The case where G = V4 o A4 yields examples of irreducible monomial fourdimensional representations of orthogonal type, which are modular by [Ra2]. However in the case where G = V4 · A4 , we obtain below irreducible monomial representations ρ which are not of orthogonal type, whence not essentially self-dual. Then ρ is not a tensor product of two two-dimensional representations since its image does not lie in GO4 (C). Nor is ρ a symmetric cube lift of a two-dimensional representation because G is not a subgroup of PGL2 (C). Proof. Let L be the field cut out by ρ and let E be the subfield corresponding to a normal subgroup of Gal(L/F ) of index 3. By definition, ρ is irreducible. So Proposition 7.4 implies that either G = V4 o A4 or V4 · A4 . Both of these groups have no elements of order 6, hence by Proposition 4.2 we only need to show that ρE and Λ2 (ρ) are modular. As remarked in the proof of Proposition 7.4, Gal(L/E) is a direct product of a 2-group P2 with a cyclic group C of odd order. Therefore Gal(L/E) is nilpotent. By a theorem of Arthur and Clozel (Proposition 9.1), this implies that ρE is modular. We now show Λ2 (ρ) is modular. We may assume that Λ2 (ρ) does not contain any characters by Proposition 7.2. Thus Λ2 (ρ) cannot contain an irreducible five-dimensional 59 representation either. Any two-dimensional representation contained in Λ2 (ρ) is modular by Theorem 3.5. Now suppose Λ2 (ρ) contains an irreducible representation τ of dimension 3 or 6. We know that all irreducible representations of Gal(L/E) have dimension a power of two because Gal(L/E) = P2 × C. Thus τE must be reducible, whence τ is induced from the normal extension E and therefore modular. Finally, consider the case where Λ2 (ρ) contains an irreducible four-dimensional representation σ. As noted in the proof of Lemma 7.2, there is a natural symmetric pairing Λ2 (ρ) × Λ2 (ρ) → Λ4 (ρ) = det(ρ). Hence Λ2 (ρ), and also σ, maps into GO6 (C). The dimension of σ implies that its image in fact lies in GO4 (C). Hence σ is modular by [Ra2]. Thus all irreducible components of Λ2 (ρ) must be modular, so Λ2 (ρ) is also, so ρ is also. To see that there are infinitely many non-essentially-self-dual hypertetrahedral representations, take the group G192 of order 192 generated by  −1    0    0  0 0 0 0   −1 0     −1 0 0   0 ,   0 1 0   0   0 0 1 0 0 0   0 0 −1 0     1 0 0   0 0 ,   0 −1 0   i 0   0 0 1 0 i 0 0 0   0 −1       0 1  , and     1 0    0 0 0 0 0 0 0    −1 0  .  0 0   0 1 As this is solvable, it occurs as a Galois group over Q by a theorem of Shafarevich ([NSW]). The group G192 has center hiIi of order 4, and G192 = G192 /hiIi ≃ V4 · A4 . This group (realized as a Galois group) has hypertetrahedral representation ρ which is not essentially self-dual and not induced from a normal extension. Such examples exist of orders 192 · k, k = 1, 2, 3, . . .. This can easily be seen by taking central products of G192 with cyclic groups. 60 Chapter 8 Three-Dimensional Representations Let ρ : GF → GL3 (C) be a Galois representation. By Theorems 3.7 and 3.8, any imprimitive (and hence monomial) three-dimensional Galois representation is modular. In particular, if ρ is essentially self-dual of solvable type then ρ is modular (cf. Proposition 1.3 of [Ra2] or Section 8.1 below). Here we will consider the case when ρ is primitive, about which little is known regarding modularity. No primitive examples of solvable type are known to be modular. In fact, the only primitive cases known are when ρ is a twist of the symmetric square of a modular two-dimensional icosahedral representation. However, for two primitive cases of solvable type, we show that the adjoint Ad(ρ) is modular. 8.1 Primitive Three-Dimensional Representations Let ρ : GF → GL3 (C) be a continuous irreducible primitive representation with projective image G. Then G is a finite subgroup of PGL3 (C). The finite subgroups of PGL3 (C) have been classified by Blichfeldt (see [MBD] or [Bl]). The primitivity condition leaves us with six possibilities for G. Fix a primitive ninth root of unity ζ9 and set ω = ζ96 . Consider the following elements of GL3 (C):     0 1 0 1 0 0         S = 0 ω 0  , T = 0 0 1 ,     0 0 ω2 1 0 0 61  ε 0 0   1 1 1      1    2 U = 0 ε 0  , V =  . 1 ω ω ω − ω2     2 0 0 εω 1 ω ω Let G36 = hS, T, V i = hS, T, V i/ ∼ ≤ PGL3 (C), G72 = hS, T, V, U V U −1 i ≤ PGL3 (C), G216 = hS, T, V, U i ≤ PGL3 (C). As is suggested from the notation G36 has order 36, G72 has order 72, and so on and so forth. Then for a primitive (irreducible) subgroup G of GL3 (C), the projective image G in PGL3 (C) falls into one of the following six cases: (1) G ≃ G36 ; (2) G ≃ G72 ; (3) G ≃ G216 ; (4) G ≃ A5 ≃ PSL2 (F3 ); (5) G ≃ A6 ; (6) G ≃ PSL2 (F7 ). The first three cases are solvable and the latter three are simple and non-solvable. Observe that case (4) is always essentially self-dual, which we shall justify presently. This means if G is a finite primitive subgroup of GL3 (C) such that G ≃ A5 , then G ⊆ GO3 (C). Let ρ be the standard representation of G. Then by Section 3 of [Ra2], ρ ≃ Sym2 (σ) ⊗ χ for some two-dimensional representation σ and some character χ. By the symmetric square lift of Gelbart and Jacquet [GeJ], the question of modularity in case (4) reduces to the modularity of two-dimensional icosahedral representations σ. However, none of the cases (1) – (3), (5) or (6) are essentially self-dual, and modularity is not known in these cases. Let G be a finite primitive subgroup of GL3 (C) with G ≃ A5 . Let us briefly explain why G ⊆ GO3 (C). Put ζm = e2πi/m . Let n be the order of G and Z = hζ3n i. Set G0 = ZG. −k Given any g ∈ G, det g = ζnk for some k. Let z = ζ3n . Then  3 −k det zg = ζ3n det g = ζn−k ζnk = 1, (8.1) 62 i.e., zg ∈ SL3 (C). Let H = G0 ∩ SL3 (C). Note that H = G0 = G ≃ A5 . Then any g ∈ G can be written as g = zh where z ∈ Z and h ∈ H. Suppose now that H ⊆ O3 (C). Let f denote the standard orthogonal form on C3 . Write any g ∈ G as g = zh as above. Then for x, y ∈ C3 , f (g(x), g(y)) = f (zh(x), zh(y)) = z 2 f (h(x), h(y)) = z 2 f (x, y), (8.2) i.e., g ∈ GO3 (C). So it suffices to show that indeed H ⊆ O3 (C). We know H is a subgroup of SL3 (C) with projective image A5 . As the center of SL3 (C) is C3 , H must either be A5 or an cyclic central extension of A5 by C3 . However the only extension of A5 by C3 is the trivial one, C3 × A5 , where C3 is embedded as the group of scalar matrices of order 3 inside SL3 (C). Thus C3 ⊆ O3 (C). So it suffices to check that A5 get embedded into O3 (C). But A5 has only two three-dimensional representations, both of which are orthogonal. Quod erat demonstrandum. 8.2 Ad : GL3 → GL8 Let ρ : GF → GL3 (C) be an irreducible representation with image G and projective image G. Then ρ⊗ ρ̌ is a nine-dimensional representation which contains the trivial representation. Write ρ ⊗ ρ̌ = 1 ⊕ Ad(ρ), (8.3) where Ad(ρ) is an eight-dimensional representation called the adjoint representation of ρ. Observe that Ad(ρ) is self-dual because ρ ⊗ ρ̌ is. Note that if ρ(g) = zI3 for some z ∈ C, then (ρ ⊗ ρ̌)(g) = zI3 ⊗ z −1 I3 = I9 . (Here Im is the m × m identity matrix.) Hence Ad(ρ) is an eight-dimensional representation which factors through G ⊆ PGL3 (C). This will be a key point in studying Ad(ρ). Proposition 8.1. Let ρ : GF → GL3 (C) be irreducible with projective image G ≃ G36 or G72 . In the former case, Ad(ρ) is a sum of two irreducible four-dimensionals representations; in the latter, Ad(ρ) is irreducible. In both cases, Ad(ρ) is induced by characters along a series of quadratic extensions. Hence Ad(ρ) is modular. Proof. First consider the case G ≃ G36 . Looking at the character table for G36 , we see that G36 has four one-dimensional representations and two irreducible four-dimensional 63 representations. Because ρ is irreducible, Ad(ρ) cannot contain any one-dimensional representations by Proposition 2.2. Hence Ad(ρ) decomposes as the sum of two four-dimensional representations. Let σ be either of the four-dimensional components of Ad(ρ). Note that G36 has a normal subgroup G18 of index 2, which itself has a normal subgroup C32 of index 2. In fact, C32 E G36 . Restricting the four-dimensional representation σ to a subgroup of order 9 must yield a sum of characters. Hence σ is induced from the normal subgroup C32 (and also from G36 ). By cyclic automorphic induction (Theorem 3.7), σ is modular. Therefore Ad(ρ) is also modular. Now consider the case G ≃ G72 . Looking at the character table for G72 , we see that G72 has four one-dimensional representations, one irreducible two-dimensional and one irreducible eight-dimensional. It is immediate from the definitions that G72 contains G36 . By the previous case, when we restrict Ad(ρ) to G72 we must get a sum of two irreducible four-dimensional representations. Thus Ad(ρ) must be irreducible and induced from the four-dimensional representations in the G36 case. Theorem 3.7 again implies that Ad(ρ) is modular. We remark that in the case G ≃ G216 , Ad(ρ) is irreducible but monomial. However it is only induced from non-normal extensions of degrees 4 and 8. In fact, G216 has no normal subgroups of index 2, 4 or 8. Suppose now that G ≃ A5 . Then ρ ≃ Sym2 (σ)⊗χ for some two-dimensional icosahedral representation σ and a character χ (cf. Section 3 of [Ra2]). Because the representation Ad : GL3 (C) → GL8 (C) factors through PGL3 (C), Ad(ρ) factors through Ad(ρ). Thus, to study Ad(ρ) it suffices to consider when ρ is actually a representation of A5 . In this case, one easily checks that Ad(ρ) ≃ ρ ⊕ Sym4 (σ) ≃ (Sym2 (σ) ⊗ χ) ⊕ Sym4 (σ). (8.4) Many cases of icosahedral representations are known to be modular. If σ is modular, then Ad(ρ) also is by the functoriality of Sym2 and Sym4 on GL2 . If G ≃ A6 , then Ad(ρ) is irreducible and primitive. Irreducibility can be seen just by looking at dimensions of the irreducible representations for A6 , which has 1 one-dimensional, 2 five-dimensional, 2 eight-dimensional, 1 nine-dimensional and 1 ten-dimensional irre- 64 ducible representations. As Ad(ρ) is eight-dimensional and does not contain the trivial representation, it must be irreducible. Primitivity follows because A6 has no subgroups of index 2, 4 or 8. If G ≃ PSL2 (F7 ), then Ad(ρ) is irreducible and monomial. Here also, irreducibility follows immediately from the character table for PSL2 (F7 ), which has 1 one-dimensional, 2 three-dimensional, 1 six-dimensional, 1 seven-dimensional and 1 eight-dimensional irreducible representations. The group PSL2 (F7 ) has no subgroups of index 2 or 4, but it does have a maximal subgroup of index 4, being the non-abelian group G21 of order 21. Restricting the eight-dimensional representation of PSL2 (F7 ) to G21 we see Ad(ρ) is monomial. On a final note, even though knowing modularity of Ad(σ) combined with a base change is enough to prove modularity for two-dimensional tetrahedral σ, it will not suffice in the three-dimensional case. One issue is that no subgroups (normal or non-normal) of index 3 exist in G36 . In fact, every maximal subgroup of G36 has index either 2 or 9. 65 Chapter 9 Supersolvable Representations A group is supersolvable if it has a finite normal series with each factor cyclic. Recall we have the following inclusions of classes of finite groups: {nilpotent groups} ⊆ {supersolvable groups} ⊆ {monomial groups} ⊆ {solvable groups} . Artin’s conjecture is known for all representations whose image is monomial and the strong Artin conjecture is known for all representations whose image is nilpotent. It is natural to ask what can be said in the case of representations with supersolvable image. We refer the reader to [Ha] for the basics on supersolvable groups. The key result we need from group theory is that any maximal subgroup of a supersolvable group has prime index. (In fact, Huppert proved that a finite group G is supersolvable if and only if all maximal subgroups are of prime index.) Let us first elaborate on the nilpotent case because somewhat more is known. Here we will use the term character for the character of any finite-dimensional representation, not just a linear character. Let G be a finite group. Recall that H is a subnormal subgroup of G if there exist subgroups H1 , H2 , . . . , Hr of G such that H E H1 E H2 E · · · E Hr E G. An accessible character of G is a Z-linear combination of characters induced from one-dimensional representations of subnormal subgroups of G. Call a representation accessible if its character is. An immediate consequence of the base change theory for GLn developed by Arthur and Clozel is Proposition 9.1. (Proposition 7.2 of [AC]) Let ρ be a continuous Galois representation over C of solvable type. If ρ is accessible, then it is modular. 66 In particular, when ρ has nilpotent image, ρ is modular. Dade [Da] showed that, if G is solvable and ρ is an irreducible accessible representation, then ρ is monomial and induced from a one-dimensional representation of a subnormal subgroup. Hence Proposition 9.1 gives us no new instances of Artin’s conjecture, though modularity is stronger and was previously unknown. Call a finite solvable group G accessible if every irreducible representation of G has an accessible character. Then the class of accessible groups also lies strictly between the nilpotent groups and the solvable groups, and has a non-trivial intersection with the class of supersolvable groups, without either class contained in the other. (For example, S3 is accessible and supersolvable but not nilpotent; A4 is accessible but not supersolvable; the group G108 below is supersolvable but not accessible.) We show modularity for certain representations with supersolvable image, some of which are accessible and some of which are not. Proposition 9.2. Let ρ : GF → GLn (C) be a continuous representation with supersolvable image. If n = 2j or n = 2j · 3 for some j ≥ 0, then ρ is modular. We need to know the following fact. Lemma 9.1. Let G be a supersolvable subgroup of S6 and H ≤ G of index 6. Then H is contained in a subgroup of G of index 2. Proof. Let G be such a group. Then |G| = 6m. The proof is to simply look at all possibilities for G and check they satisfy the desired property. We will omit the details, which can easily be checked in [GAP] for instance. If m = 1, G = C6 or G = S3 . In both cases the lemma obviously holds. If m = 2, G = D12 and it is easy to see the lemma holds in this case also. If m = 3, G = C3 × S3 or G = C3 · S3 = h(1, 2, 3), (4, 5, 6), (1, 2)(4, 5)i. If m = 6, then G = S3 × S3 . There are no other supersolvable subgroups of S6 of order divisible by 6. Proof of Proposition. Let L be the field cut out by ρ and G = Gal(L/F ). We view ρ as a representation of G. As G is supersolvable, ρ is monomial. Say ρ is induced from a character χ of Gal(L/K) for some extension K/F of degree n. When j = 0 this is either the well-known one-dimensional case or the case of cubic induction from GL1 to GL3 (Theorems 3.7 and 3.8). Suppose j ≥ 1. 67 First consider the case n = 2j . Let M be the field corresponding to a maximal subgroup of G containing Gal(L/K). Since G is supersolvable, [M : F ] = p for some prime p. Let F n σ = IndM K χ. So σ is p -dimensional and ρ = IndM σ. In particular, p|n. If M is normal and σ is modular, then so is ρ by automorphic induction (Theorem 3.7). As n = 2, we know p = 2 and M is normal. Hence, by Fermat’s method of descent, we can reduce to the j = 0 case and obtain that ρ is modular. e be the Galois closure of K/F . Then the factor group Now assume n = 2j · 3. Let K e ) of G is supersolvable, so there exists a maximal subgroup H1 of prime index in Gal(K/F e ) which contains Gal(K/K). e Gal(K/F Similarly there exists a subgroup H2 of prime index e in H1 which contains Gal(K/K). Repeating this, we get a maximal series of groups e e ), Gal(K/K) = Hj+1 ≤ Hj ≤ · · · ≤ H1 ≤ H0 = Gal(K/F (9.1) each of prime index in the next. Then we have a corresponding tower of fields K = Kj+1 ⊇ Kj ⊇ · · · ⊇ K1 ⊇ K0 = F. (9.2) One of the terms, say Ki+1 ⊇ Ki , must be index 3 and the rest are index 2. We want to reduce to the case where i = j, so assume that i < j. For convenience, let us e be the Galois closure of N/K. Therefore write N = Ki+2 , M = Ki+1 , and E = Ki . Let N e /E) is a subgroup of S6 . Note that Gal(N e /E) is supersolvable because Gal(L/E) Gal(N e /N ) is contained in a subgroup of index 2. Let M 0 be the is. By the lemma above, Gal(N corresponding field extension so [N : M 0 ] = 3 and [M 0 : E] = 2. Pictorially we have the following diagram. e N NC ≤S6 2 M 3 E CC CC3 CC C M0 { {{ {{2 { {{ Thus we can reduce to the case where [Ki+2 : Ki+1 ] = 3 and [Ki+1 : Ki ] = 2. Repeating this process allows us to assume that i = j as desired. In other words, we have a tower of 68 fields 3 2 2 2 K = Kj+1 ⊇ Kj ⊇ · · · ⊇ K1 ⊇ K0 = F. (9.3) K Then IndKj χ is modular by the j = 0 case. As all of the subsequent extensions Ki /Ki−1 are normal for j ≥ i ≥ 1, we can apply normal (quadratic) automorphic induction j times   K to say that ρ = IndFK χ = IndFKj IndKj χ is modular. Remarks. Suppose ρ : GF → GLn (C) has finite supersolvable image. (1) If n = 2j , then ρ is accessible and modularity was known by Arthur and Clozel. However, ρ need not have nilpotent image, e.g., n = 2 and Im(ρ) = S3 . Note that even for n = 2 there exist non-accessible representations of solvable type, namely the ones which are primitive, i.e., the tetrahedral and octahedral representations. (2) If n = 2j · 3, then whether ρ is accessible will depend upon whether the index 3 extension Ki+1 /Ki can be chosen to be normal. For example, there exists a supersolvable group G108 of order 108 which is a semidirect product G108 = C32 oD12 and has an irreducible six-dimensional representation ρ. The group G108 has no normal subgroups of index 3. It has three subgroups H of index 2, and ρH is only reducible for one of them, G54 = C32 o S3 . By the proof of the proposition above, ρ is induced from a subgroup of index 3 of G54 , but none of these subgroups are normal. Hence ρ is not accessible. (Note: the semidirect product notation does not determine G54 or G108 uniquely, but the GAP notation given in Appendix B does.) 69 Appendix α Here we give a proof of the following result, shown to us by Ramakrishnan. It is very similar to the proof of Proposition 3.2.1 in [Ra], and the ideas for GL2 go back to [DS]. Proposition A.1. Let F be a number field and let σ : GF → GLn (C) be an irreducible continuous Galois representation. Let π be a cuspidal automorphic representation of GLn (AF ) such that L(s, πv ) = L(s, σv ) for almost all v. Then L(s, πv ) = L(s, σv ) for all finite Q Q places v. Moreover, v|∞ L(s, πv ) = v|∞ L(s, σv ). In other words, we have the equalities L(s, π) = L∗ (s, σ) of Euler products over Q and L(s, πf ) = L(s, σ) of Euler products over F , where πf = ⊗0v<∞ πv . Note that when F = Q, this proposition says all of the local L-factors are equal. When F is CM or totally real, a similar result at the finite places is contained in Section 6 of [He]. Proof. Let S designate a finite set of places containing the ramified places such that L(s, πv ) = L(s, σv ) for all v 6∈ S. We first do the case of infinite primes. By [JPSS2], we may choose a character χ of F such that χ∞ = 1 and χu is sufficiently ramified so that −1 L(s, πu ⊗ χu ) = L(s, π̌u ⊗ χ−1 u ) = L(s, σu ⊗ χu ) = L(s, σ̌u ⊗ χu ) = 1 (A.1) for all u ∈ S. Then L(s, πf ⊗ χf ) = L(s, σ ⊗ χ) and L(s, π̌f ⊗ χf ) = L(s, σ̌ ⊗ χ), where χ = χf χ∞ . Dividing the functional equation for L(s, π ⊗ χ) by that for L∗ (s, σ ⊗ χ) gives ε(s, π ⊗ χ) L(s, π ⊗ χ) L(1 − s, π̌ ⊗ χ−1 ) = . ∗ −1 L (1 − s, σ̌ ⊗ χ ) ε(s, σ ⊗ χ) L∗ (s, σ ⊗ χ) (A.2) Set L∞ (s, π) = Q 70 Q v|∞ L(s, πv ) and L∞ (s, σ) = v|∞ L(s, σv ). We have L(s, π ⊗ χ) = L∞ (s, π ⊗ χ)L(s, πf ⊗ χf ) = L∞ (s, π)L(s, πf ⊗ χf ), (A.3) L∗ (s, σ ⊗ χ) = L∞ (s, σ ⊗ χ)L(s, σ ⊗ χ) = L∞ (s, σ)L(s, σ ⊗ χ), (A.4) and similarly for π̌ and σ̌. Using this, (A.2) becomes L∞ (1 − s, π̌) L∞ (s, π) = ε(s) , L∞ (1 − s, σ̌) L∞ (s, σ) (A.5) where ε(s) = ε(s,π⊗χ) ε(s,σ⊗χ) is a non-vanishing entire function. Because π is cuspidal, a result of Shalika [Shal] tells us that πv is generic for all v. Then by Proposition 2.1 of [BR], L(s, πv ) and L(s, π̌v ) have no poles in <(s) > 21 − δ for some δ. Also L∞ (s, σ) and L∞ (s, σ̌) have no poles to the right of <(s) = 0; this is because σ has finite image by assumption, thus making the Frobenius eigenvalues roots of unity. In other words, no pole of L∞ (s, π) is also a pole of L∞ (1 − s, π̌) and no pole of L∞ (s, σ) is a pole of L∞ (1 − s, σ̌). Since the local factors are never zero, the poles of L∞ (s, π) must coincide with the poles of L∞ (s, σ), with the right multiplicities. We can write n[F :Q] n[F :Q] L∞ (s, π) = Y ΓR (s + aj ), L∞ (s, σ) = Y ΓR (s + bj ), (A.6) 1 1 where ΓR (s) = π −s/2 Γ(s/2). Because their poles agree, we must have {aj } = {bj }, i.e., L∞ (s, π) = L∞ (s, σ). Now fix a finite place v ∈ S. Choose a character χ such that χv = 1 and (A.1) holds for each u ∈ S − {v}. By the method above, we get L(1 − s, π̌v ) L(s, πv ) = ε(s) , L(1 − s, σ̌v ) L(s, σv ) (A.7) where ε(s) = ε(s,π⊗χ) ε(s,σ⊗χ) is a non-vanishing entire function. As above, using Proposition 2.1 of [BR], no pole of L(s, πv ) is also a pole of L(1 − s, π̌v ). Similarly no pole of L(s, σv ) is a pole of L(1 − s, σ̌v ). Hence the poles of L(s, πv ) must coincide with the poles of L(s, σv ), and therefore L(s, πv ) = L(s, σv ). 71 Appendix B GAP Notation The following table provides GAP Notation for many groups appearing in the text. The group [a, b] is the group of order a accessed by the command SmallGroup(a, b) in [GAP]. 72 G [a, b] F20 [20,3] Q28 [32, 49] Q8 D8 [32, 50] Q28 C4 [64, 266] Out(Q28 ) = O+ 4 (F2 ) [72, 40] E24 · C5 [80, 49] E24 · D10 [160, 234] Q8 D8 · C5 [160, 199] Q8 D8 · D10 [320, 1581] Out(Q28 C4 ) = Sp4 (F2 ) × C2 [1440, 5842] V 4 · A4 [48, 3] C4 × A4 [48, 31] SL2 (F3 ) × C2 [48, 32] SL2 (F3 ) o C2 [48, 33] V 4 × A4 [48, 49] V 4 o A4 [48, 50] G192 [192, 4] G36 [36, 9] G72 [72, 41] G216 [216, 153] G54 [54, 8] G108 [108, 17] 73 References [AC] Arthur, James; Clozel, Laurent. 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