Images of Adelic Representations of Modular Forms

Transcription

1 MSc Mathematics Master Thesis Images of Adelic Representations of Modular Forms Author: Robert Mann Supervisor: dr. A.L. Kret Examination date: August 7, 2018 Korteweg-de Vries Institute for Mathematics

2 Abstract Serre was able to prove in [18] that the image of adelic representation of elliptic curves without complex multiplication are open. In this thesis we explain and give proofs of the main results of David Loeffler s paper Images of adelic Galois representations for modular forms[8], wherein he gives an analogue of Serre s results for modular forms without complex multiplication. He is then able to generalise this result to finite products of representations of modular forms without complex multiplication, subject to certain conditions. Title: Images of Adelic Representations of Modular Forms Author: Robert Mann, mann.robertj@gmail.com, Supervisor: dr. A.L. Kret Second Examiner: prof. dr. L.D.J. Taelman Examination date: August 7, 2018 Korteweg-de Vries Institute for Mathematics University of Amsterdam Science Park , 1098 XG Amsterdam 2

3 Contents Introduction 4 1 Preliminaries The absolute Galois group Elliptic Curves Modular Forms Galois representations of Modular Forms Why study Galois representations of modular forms and elliptic curves? The modularity theorem Fermat s last theorem Images of adelic representations of modular forms Is the image open? Inner Twists The quaternion algebra B Constructing B Properties of B Image of H Joint large image Popular summary 43 3

4 Introduction In loose terms 1, modular forms of weight k and level N are holomorphic functions on the upper-half plane that have symmetry properties relating to an integer k 2 and the congruence subgroup ( ) a b a d 1 (mod N) Γ 1 (N) = SL c d 2 (Z) : c 0 (mod N). As such, they are analytic objects, but they also heavily connected to number theory via the Langlands program. Crucially the set of modular forms for a chosen pair k 2, N of integers forms a finite dimensional vector space M k (N) over C. This vector space can be decomposed into the space of modular forms that arise from a smaller choice of N, and the space of newforms. A chosen newform f (that satisfies some extra conditions) will give rise to a number field L, and the adelic Galois representation of f, ρ f : G Q GL 2 (L Q A Q ), where G Q is the absolute Galois group of Q, and A Q denotes the finite adèle ring of Q. These representations are constructed from the Tate modules of abelian varieties Q whose sets of C-rational points are of the form C g /Z 2g note that these varieties do depend on the choice of embedding Z 2g C g and so they can be viewed as higher dimensional analogues of those constructed from elliptic curves where g = 1. In 1995 Andrew Wiles was able to prove that the representations of semi-stable elliptic curves are given by modular forms with k = 2, and in doing so he was able to prove Fermat s last theorem. Later in 2001 it was proven that the same result holds for all elliptic curves, and so representations of modular forms can be seen as a generalisation of those for elliptic curves. However the representations of modular forms are in some ways less well-behaved than those of elliptic curves. In particular ρ f typically does not have open image, whereas Serre proved in 1972 that representations of elliptic curves without complex multiplication always have open image. In [10], Momose constructed an open subgroup H of G Q, a subextension F of L/Q, and a quaternion algebra B over F such that for each prime number l, the l-adic representation ρ f,l H has open image inside the subgroup of elements (B Q Q l ) with norm inside Q (k 1) l, where Q (k 1) l denotes the group of (k 1)-th powers inside Q l. 1 Full definitions are saved for the chapter Preliminaries. 4

5 Ribet then built on this result in [16], demonstrating that for all but finitely many primes l, B Q Q l = M 2 (F Q Q l ), and the image of ρ f,l H is conjugate to the subgroup of GL 2 (O F Z Z l with determinant a (k 1)-th power in Z l. After seeing these results one would expect that the image of ρ f H inside {x (B Q A Q ) : det x Z (k 1) l } would also be open. However this result does not follow immediately for a fairly simple group-theoretic reason: the group ρ f (H) does not necessarily contain ρ f,l (H), where ρ f,l (H) is identified with p l {1} ρ f,l(h) inside (B Q A Q ). Loeffler was able to demonstrate in [8] that ρ f H χ where χ is the cyclotomic character has open image in the group {(x, λ) (B Q A Q ) A Q : norm B/F = λ k 1 }. Now let {f 1,..., f n } be a finite set of modular forms, and denote the weight of f i by k i. Denote the quaternion algebra corresponding to f i by B i. Assume also that the f i are pairwise non-conjugate (see the section Inner Twists ). Then there exists an open subgroup H of G Q, such that the image of (ρ f1 ρ fn ) H χ is open inside n (x 1,..., x n, λ) (B i Q A Q ) A Q : det x i = λ k i 1. i=1 5

6 1 Preliminaries We begin by defining modular forms and their adelic Galois representations. Also, as a precursor to the main topic of this thesis we will discuss Galois representations of elliptic curves. The reader who is familiar with these concepts can safely skip this chapter. 1.1 The absolute Galois group The Galois representations we are interested in will always be defined on the Galois group G Q = Gal(Q/Q), and so the purpose of this chapter is to give this group the structure of a topological group, to discuss how Galois theory applies to G Q, and to explain what Galois representations on G Q are. Let L/F be a Galois field extension. The case where L/F is of infinite degree is more complicated than the finite degree case for instance subgroups of Gal(L/F ) do not correspond to intermediate fields. To have a Galois theory of infinite extensions we first need to give Gal(L/F ) the Krull topology, which is the group topology with basis sets around the identity of the form Gal(L/K), where K/F is a field extension of finite degree. Equivalently, we may describe Gal(L/F ) and the Krull topology as the profinite group given by the projective limit, lim Gal(K/F ), K with the usual profinite topology, where K ranges over the finite Galois sub-extensions of L/F. The intermediate fields of L/F then correspond to closed subgroups of Gal(L/F ), and Galois sub-extensions to normal closed subgroups, in the usual inclusion-reversing manner. If K is a field with separable field closure K sep, then the absolute Galois group of K is G K = Gal(K sep /K). Given an algebraic field extension F/Q, that is possibly infinite, we write O F for the integer ring of F, and in the case F = Q we simply write O = O Q. If p is a maximal ideal of O F then we write k(p) = O F /p for the residue field of p. A finite place of F is a p-adic norm, where p is a maximal ideals of O F, and the infinite places of F are the field norms given by embeddings F C. If an infinite place of F is given by an embedding F R, then we call that place a real place of F, and otherwise we call it a complex place of F. Note that complex places of F are always given by pairs of complex conjugate embeddings F C. We also refer to finite places of F as finite primes of F, and infinite places of F as infinite primes of F, and so on. If L/F is an algebraic field extension with w a place of L and v a place of F, then w is said to lie above v if a field norm corresponding to w can be restricted to a field norm corresponding to v. If w is a finite place then this means w is given by a finite prime P 6

7 of L such that P F = p is the finite prime of F corresponding to v. When we refer to finite primes of F we will usually write p, P, or λ, unless if F = Q in which case we write p or l. When we refer to places (finite or infinite) of F we will use v or w. If v is the finite place corresponding to a prime p then we use F v and F p interchangeably, as well as k(v) and k(p). Assume now that F is a number field and K/F is a (possibly infinite) Galois extension. Let P be a finite prime of K, p P a finite prime of F, and p p a prime number. Then k(p) = F q for some prime power q = p n, and k(p)/k(p) is a Galois extension. The decomposition group D P of P is the subgroup of Gal(K/F ) fixing P. Now, because D P fixes P and F, it acts on k(p) in a way that fixes k(p), hence we have a surjection D P Gal(k(P)/k(p)), whose kernel I P is called the inertia group of P. The group Gal(k(P)/k(p)) is either finite cyclic or isomorphic to the profinite integers Ẑ. In either case it is topologically generated by the element σ q : x x q, called the Frobenius element of Gal(k(P)/k(p)). Any element of D P mapping to σ q is called a Frobenius element of P, and is denoted by Frob P. Assume now that p is unramified in K, which means that the inertia group of P is trivial, then the Frobenius element Frob P is unique. Moreover if σ Gal(K/F ) then σp is also a finite prime lying over p, and it follows that Frob σp = σ Frob P σ 1. In this case the Frobenius elements of finite primes lying over p form a conjugacy class, so we may define Frob p to be the Frobenius element of some finite prime P lying over p, and then Frob p is defined up to conjugacy. We will now see that the Frobenius elements in G Q are dense; in fact we will see that a stronger statement is true. Theorem (Chebotarev s density theorem). Let K/F be a finite Galois extension of number fields of degree n, and let X be a union of conjugacy classes in Gal(K/F ), then the set of finite primes p of F whose Frobenius elements lie in X has density #X/n. Corollary Let P be a set containing all but finitely many prime numbers, and for each p P and each finite prime p of Q lying over p pick a Frobenius element Frob p, then these Frobenius elements are dense in G Q. Definition Let l be a prime number. An l-adic representation of Gal(K/F ) is a continuous group homomorphism ρ : Gal(K/F ) GL n (Q l ) for some positive integer n. The representation ρ is called unramified at a finite prime p of F if ρ is trivial on the inertia groups over p. Because Gal(K/F ) is a compact group, we can show that ρ is isomorphic over Q l to a representation taking values in GL n (Z l ). First, note that each l-adic lattice in Q n l is conjugate to Z n l, and so is fixed by a subgroup of GL n(q l ) conjugate to GL n (Z l ), which is open. Now, let Λ be an l-adic lattice inside Q n l of full rank. By continuity 7

8 of ρ there exists a normal open subgroup H of Gal(K/F ) such that H fixes Λ. The subgroup H has finite index, so there exists a finite set {σ 1,..., σ j } of representatives of Gal(K/F )/H, and the set Λ = j i=1 ρ(σ i)λ is a new l-adic lattice inside Q n l that is fixed by Gal(K/F ). Hence the image of ρ is conjugate to a subgroup of GL n (Z l ). For each prime number p, let Frob p G Q be a Frobenius element of a finite prime over p. Let ρ and τ be two semi-simple l-adic representations of G Q, and assume there exists N 1 such that ρ and τ are unramified at all primes p N. If Tr(ρ(Frob p )) = Tr(τ(Frob p )) for all primes p N, then by corollary , the representations ρ and τ have the same character, and hence are conjugate as representations over Q l. 1.2 Elliptic Curves In this section we discuss elliptic curves over Q, their Galois representations, and give the results proven by Serre in his work during the late sixties and early seventies. The very short version is that elliptic curves over Q can be seen as 2-dimensional tori C/Z 2, and we can use these tori to construct Galois representations in a natural way. Later we will see that representations of modular forms are constructed from higher dimensional tori C n /Z 2n (n 1) in a similar way, and so in this way the representations of modular forms can be seen as a higher dimensional analogue of the representations of elliptic curves. And in fact due to the modularity theorem they are a generalisation. An elliptic curve E over a field k is a non-singular projective k-variety defined by a cubic equation of the form E : Y 2 Z + a 1 XY Z + a 3 Y Z 2 = X 3 + a 2 X 2 Z + a 4 XZ 2 + a 6 Z 3 with a 1,..., a 6 k. This equation is called a Weierstrass equation for E. If the characteristic of k is neither 2 nor 3 then by a change of variables we may write Y 2 Z = X 3 + axz 2 + bz 3 with a, b k, such that the discriminant 16(4a b 2 ) is non-zero. Note that the point at infinity, O = (0 : 1 : 0), is always a point of inflection. Crucially we can give E the structure of an abelian variety over k. Let K be some field extension of k, and let P and Q be two distinct points on E(K), the line through P and Q intersects E(K) in a third point, denoted P Q. We then define P + Q = O(P Q). When P = Q we simply use the tangent line, and take P P to be the third point where the tangent line intersects E. Finally we have O + O = O. Because these operations are all defined by polynomials over k, we find that + gives E the structure of an abelian variety over k. For a more detailed argument the reader should consult [9, I.3] Assume now that E is defined over C. We can use the Weierstrass -function to find a biholomorphic group isomorphism E(C) C/Λ, with Λ a full lattice in C. Given 8

9 τ h, with h the upper-half complex plane, we write Λ = Z + τz, (z; τ) = 1 z 2 + ( 1 (z + w) 2 1 ) w 2, and g 2 = 60 w Λ\{0} g 3 = 140 w Λ\{0} w Λ\{0} w 4, w 6. One can then show by comparing the poles at 0 that (z; τ) 2 = 4 (z; τ) 3 g 2 (z; τ) g 3, where τ is treated as a constant, and the differentiation is with respect to z. We then find that the function C C C given by z ( (z; τ), [ (z; τ)/2] 2 ) gives a biholomorphic group isomorphism between C/Λ and the C-rational points of the elliptic curve E : Y 2 Z = X 3 g 2 4 XZ2 g 3 4 Z3. In this way we find that E(C) is always a torus. For the remainder of this section we assume that E is defined over Q. Now, E can also be studied through its torsion groups, defined by E[N] = x E(C) : N x = x + + x }{{} N = O, N Z 1. Because the group structure is given by polynomials over Q, the torsion points are all Q-rational points. Taking all the torsion points over all N we get a dense subset of E, and so by taking the projective limit of the torsion groups we may study all these points simultaneously. We call this projective limit the Tate module, and denote it by T (E). Additionally, using the isomorphism E(C) = C/Z 2, we find that which gives E[N] = Z/NZ Z/NZ, T (E) = Ẑ Ẑ. Finally we define an action of G Q on E(Q) via σ(x : y : z) = (x σ, y σ, z σ ). Because the abelian group structure on E is given by polynomials over Q, this action preserves 9

10 addition, and so it acts on each torsion group E(N), and hence the Tate module T (E). The result is that we have a representation ρ E : G Q GL 2 (Ẑ) that is well-defined up to conjugacy. To see that ρ E is continuous, consider U = 1 + N M 2 (Ẑ). The preimage of U under ρ E is simply Gal(Q/Q(E[N])), which is open. The image of ρ E depends largely on the structure of the ring R of Q-endomorphisms of E, or to be more precise if the natural inclusion Z R is an isomorphism or not. If we have Z = R then we say that E has complex multiplication. To see what other endomorphisms are possible we consider endomorphisms of E(C) viewed as a torus. Any non-trivial endomorphism of E(C) is given by multiplication by some non-zero λ C satisfying λλ Λ. Let R be the ring of endomorphisms of E. First, each member of Z gives an endomorphism of E, so Z R. If there exists α / R \ Z, then by considering the action of α on ω 1 and ω 2 we find that ω 1 /ω 2 satisfies a quadratic equation over Z, and so Q(ω 1 /ω 2 ) is an imaginary quadratic field, and so R is an order inside Q(ω 1 /ω 2 ). Finally we come to Serre s result for Galois representations of elliptic curves: Theorem Let E be an elliptic curve over Q without complex multiplication, then the image of ρ E in GL 2 (Ẑ) is open. Proof. See [18, theorem 3]. For each prime number l we naturally obtain a representation ρ E,l given by projection from GL 2 (Ẑ) to GL 2(Z l ), and we can immediately see that when E is without complex multiplication, the representation ρ E,l always has open image, and is surjective for all but finitely many prime numbers l. 1.3 Modular Forms Modular forms are particularly well-behaved holomorphic functions define on the upperhalf plane which satisfy symmetry properties relating to SL 2 (Z). These modular forms are linked to number theory in the sense that many functions of interest to number theorists, such as the partition function and the divisor sum function, can be used to construct modular forms, and then those modular forms can in turn be used to prove properties about the original number theoretic object. They can also be used to construct Galois representations which are of interest to number theorists because they give us a way to study G Q. Define an action of SL 2 (Z) on P 1 (C) by γ (z : w) = (az + bw : cz + dw), γ = ( a c ) b SL d 2 (Z), (z : w) P 1 (C). 10

11 We write = (1 : 0) and identify C with a subspace of P 1 (C) via the inclusion map Then, by an abuse of notation, we write γ z = az + b cz + d, C P 1 (C); z (z : 1). γ SL 2(Z), z P 1 (C). with the understanding that if z =, then γ z = a/c, and if cz + d = 0 then γ z =. By computing the action of SL 2 (Z) on the imaginary part of a complex number we find that the action of SL 2 (Z) on C restricts to an action on the upper-half plane h = {z C : Iz > 0}, and we also find that it gives an action on P 1 (Q); we will use these two actions in particular in the definition of a modular form. We also write and j(γ, z) = cz + d (f k γ)(z) = j(γ, z) k f(γ z) where f is a complex-valued function on the upper-half plane h, and k is any integer. We may check that this gives a group action on the set of holomorphic functions h C. Definition (Congruence subgroups). Given a positive integer N, we write Γ(N) := { γ SL 2 (Z) : γ 1 (mod N) }, where 1 denotes the identity matrix. We call Γ(N) the principal congruence subgroup of level N. If Γ is a subgroup of SL 2 (Z) containing Γ(N) for some N, then Γ is called a congruence subgroup, and the level of Γ is the smallest positive integer N satisfying Γ(N) Γ. Note that Γ(N) is the kernel of the modulo N reduction map SL 2 (Z) SL 2 (Z/NZ), so Γ has finite index in SL 2 (Z). (The reduction map is surjective, but because we do not need this fact we will not prove it.) For the remainder of this section N will denote a positive integer, Γ a congruence subgroup of level N, and k an arbitrary integer. Definition (Weakly modular functions). A function f : h C is weakly modular of weight k with respect to Γ if it is meromorphic, and for all γ Γ we have f k γ = f. Definition (Cusps). Two points x, y P 1 (Q) are called Γ-equivalent if there exists γ Γ such that γ x = y. A maximal set of Γ-equivalent points in P 1 (Q) is called a cusp of Γ. Equivalently, we can define a cusp c of Γ to be a member of Γ\ SL 2 (Z)/ SL 2 (Z), where SL 2 (Z) is the subgroup of SL 2 (Z) fixing. The two definitions can be seen to be equivalent by saying that r P 1 (Q) and α SL 2 (Z)/ SL 2 (Z) represent the same cusp if and only if α is Γ-equivalent to r. We denote the set of cusps of Γ by Cusps(Γ), and we will typically use both definitions of a cusp interchangeably. 11

12 Let c be a cusp of Γ, let α SL 2 (Z) be a representative of c, and let h c be the smallest positive integer such that ( ) 1 h c α 1 Γα SL (Z). Let f be weakly modular of weight k with respect to Γ, then f k α is periodic with period h c, so by letting q c = exp(2πiz/h c ), we may define a function f k α : D \ {0} C (where D denotes the unit disk in the complex plane) that satisfies f k α(q c ) = f(z). Now f k α is holomorphic on its domain if and only if f k α is holomorphic on h, and we say f k α is holomorphic at, or that f is holomorphic at c, if f k α extends holomorphically to a function f k α : D C. When this occurs we have a Fourier series f k α(z) = a n (f k α)qc n. n=0 Additionally if a 0 (f k α) = 0, we say that f k α vanishes at, or that f vanishes at c. Definition (Modular forms). A function f : h C is a modular form of weight k with respect to Γ if it is holomorphic, weakly modular of weight k with respect Γ, and if f is holomorphic at each cusp of Γ. The set of modular forms of weight k with respect to Γ is denoted by M k (Γ), and forms a complex vector space of finite dimension. Additionally, if f also vanishes at every cusp of Γ, then f is called a cusp form of weight k with respect to Γ. The set of cusp forms of weight k with respect to Γ is denoted by S k (Γ), and forms a linear subspace of M k (Γ). Remark The direct sum M(Γ) = k Z M k (Γ), with point-wise addition and multiplication, forms a graded ring with graded ideal S(Γ) = k Z S k (Γ). In this paper we will frequently refer to the following congruence subgroups: ( ) a b Γ 0 (N) := SL c d 2 (Z) : c 0 (mod N), and ( a Γ 1 (N) := c ) b Γ d 0 (Z) : a d 1 (mod N). A member of M k (Γ 1 (N)) will be referred to as a modular form of level N and weight k. 12

13 Definition (Diamond operators). Observe that the map ( ) Γ 0 (N) (Z/NZ) a b ; d c d is a surjective homomorphism with kernel Γ 1 (N). Therefore Γ 1 (N) is a normal subgroup of Γ 0 (N) and we have Γ 0 (N)/Γ 1 (N) = (Z/NZ). It follows that we may define an action of Γ 0 (N) on M k (Γ 1 (N)) by γ f = f k γ, and so we may use the congruence Γ 0 (N)/Γ 1 (N) = (Z/NZ) to define an action of (Z/NZ) on M k (Γ 1 (N)). We call this action the diamond operator, and we denote it by d f for d (Z/NZ) and f M k (Γ 1 (N)). Definition (Dirichlet characters). A group homomorphism (Z/NZ) C is called a Dirichlet character. For each Dirichlet character χ there exists a minimal N N and a Dirichlet character χ : (Z/N Z) C such that χ is the composition (Z/NZ) (Z/N Z) χ C. We call N the conductor of χ, and χ is called primitive if N = N. Definition (Nebentypus). Let χ : (Z/NZ) C be a Dirichlet character. A modular form f M k (Γ 1 (N)) is said to have nebentypus χ if d f = χ(d)f for all d (Z/NZ). Alternatively f is said to be of character χ. Note that f S k (Γ 0 (N)) precisely when f has nebentypus 1. Definition (Hecke operators). For each prime number p we define an operator T p on M k (Γ 1 (N)) by T p f = f k β, β where β ranges over a set of representatives of ( ) 1 0 Γ 1 (N)\Γ 1 (N) Γ 0 p 1 (N). The diamond operators and the T p operators form the Hecke operators. Lemma The diamond operators and the T p operators generate a commutative Z-algebra, denoted T(M k (Γ 1 (N))), and referred to as a Hecke algebra. Proof. [3][Proposition ] In our study of newforms it will be important to have an inner-product on S k (Γ 1 (N)). This will allow us to decide which cusp forms of level N come from cusp forms of a lower level, and which are canonically of level N. 13

14 Definition (Petersson inner product). Let D = {z h : 1/2 Rz 1/2 and z 1}. Note that every point in h is either SL 2 (Z)-equivalent to two points in D, in which case those two points lie on the boundary, or just one point in D. We call D the fundamental domain. Now letting R be a system of representatives for Γ\ SL 2 (Z) we write D Γ = γ R γd. The Petersson inner product, Γ on S k (Γ) is defined by: k dx dy f, g Γ = f(z)g(z)y z D Γ y 2, where x and y are the real and imaginary parts of z respectively. Definition (Oldforms and newforms). Let M be a positive integer dividing N, and let d divide N/M. We may embed S k (Γ 1 (M)) into S k (Γ 1 (N)) via f f k α d with ( ) d 0 α =. 0 1 For each d N we let i d be the map defined by The space of oldforms of level N is i d : (S k (Γ 1 (Nd 1 ))) 2 S k (Γ 1 (N)) (f, g) f + g k α d. S k (Γ 1 (N)) old = p i p ((S k (Γ 1 (Np 1 ))) 2 ), where p ranges over the prime divisors of N, and the space of newforms of level N is the orthogonal complement with respect to the Petersson inner product: S k (Γ 1 (N)) new = (S k (Γ 1 ) old ). Definition An eigenform, sometimes called a Hecke eigenform is a non-zero modular form f M k (Γ 1 (N)) that is an eigenvector for all Hecke operators T p for p prime, and all diamond operators d with d (Z/NZ). We call f a normalised eigenform if a 1 (f) = 0. A primitive form of weight k for Γ 1 (N) is a normalised eigenform in S k (Γ 1 (N)) new. 14

15 1.4 Galois representations of Modular Forms The purpose of this section is to give a very rough sketch of the construction of the Galois representations of a normalised cuspidal eigenform of weight 2, level N, and nebentypus ε. In this thesis we will be considering representations of newforms of weight k 2, however the construction of these representations is substantially more complicated than the weight 2 case, and falls outside the scope of this thesis. The construction we give here is due to Shimura, building upon the Eichler-Shimura relation. For a full treatment of the weight 2 case the reader is directed to [3], A First Course in Modular Forms by Fred Diamond and Jerry Shurman. For a treatment of the weight 2 case, see [5, chapter 2]. Unlike in our treatment of Galois representations of elliptic curves we only look at a single prime number l, and construct a representation ρ f,l : G Q GL 2 (L Q Q l ), where L is the number field attached to f. Given a congruence subgroup Γ, one can show that the set Y (Γ) C = Γ\h with the quotient topology is a non-compact Riemann surface, see for example chapter 2 of [3]. When showing that Y (Γ) C is a Riemann surface one By adding points corresponding to the cusps of Γ we obtain a compact Riemann surface X(Γ) C. For ease of notation, we write Y (Γ 0 (N)) C = Y 0 (N) C, Y (Γ 1 (N)) C = Y 1 (N) C, and Y (Γ(N)) C = Y (N) C, as well as X(Γ 0 (N)) C = X 0 (N) C, X(Γ 1 (N)) C = X 1 (N) C, and X(Γ(N)) C = X(N) C. We can also consider X 1 (N) C to be the set of C-rational points of a variety over Q, by which we mean that there exists a projective nonsingular algebraic curve X 1 (N) over Q, such that X 1 (N)(C) with its analytic topology is isomorphic to X 1 (N) C, see [3, p.386]. We will mainly be interested in X 1 (N)(Q) and X 1 (N) C, so for ease of notation we refer to the former set as X 1 (N). First we construct an action of G Q on the l-adic Tate module of the Jacobian group J 1 (N) = Jac(X 1 (N) C ). To do this we let G Q act on the divisor group of X 1 (N) in a coordinate-wise manner, or written symbolically, σp = P σ for P X 1 (N), σ G Q. This action then generates an action on the Picard group Pic 0 (X 1 (N)), and so on the l n -torsion group Pic 0 (X 1 (N))[l n ]. This torsion group is in fact isomorphic to Pic 0 (X 1 (N) C )[l n ], see [3, p.386]. So using the isomorphism J 1 (N) = Pic(X 1 (N) C ), we have an action of G Q on T l (J 1 (N)). On the other hand J 1 (N) = Jac(X 1 (N) C ) is known to be isomorphic to a torus C g /Λ, where Λ is a full-lattice in C g. Therefore, we have an isomorphism, T l (J 1 (N)) = lim{x J 1 (N) : l n x = 0} = Z 2g n 1 l. 15

16 From this we obtain an action of G Q on the Tate module of J 1 (N), and hence a representation ρ X1 (N),l : G Q Aut(T l (J 1 (N))) = GL 2g (Z l ), which is the 2g-dimensional l-adic representation attached to X 1 (N). As with elliptic curves this representation is only defined up to conjugation. Of course this representation depends only on the level of f, so the next step is to use ρ X1 (N),l to construct a representation, does depend on f. We will now define an action of the Hecke algebra T = T(M k (Γ 1 (N))) for Γ 1 (N) on J 1 (N). Recall that the Weil pairing e N of an elliptic curve E = C/(ω 1 Z + ω 2 Z) is the function e N : E[N] E[N] µ N, defined by where γ M 2 (Z/NZ) satisfies inside E[N]. e N (P, Q) = exp(2πi det γ/n), ( ) ( ) P ω = γ 1 /N Q ω 2 /N Definition (Enhanced elliptic curves). An enhanced elliptic curve for Γ {Γ 0 (N), Γ 1 (N), Γ(N)} is an ordered pair (E, X), where E is an elliptic curve over C, and X is a point in E with order N if Γ = Γ 1 (N), a cyclic subgroup of E with order N if Γ = Γ 0 (N), or a pair (P, Q) E E with P, Q = E[N] and e N (P, Q) = e 2πi/N if Γ = Γ(N). Two enhanced elliptic curves (E, X) and (E, X ) for Γ are equivalent, denoted (E, X) (E, X ), if there exists an isomorphism of elliptic curves ϕ : E E with ϕ(x) = X. If X is an ordered pair then ϕ acts on X entry-wise. The equivalence classes of enhanced elliptic curves for Γ 0 (N), Γ 1 (N), and Γ(N) are denoted S 0 (N), S 1 (N), and S(N) respectively. It turns out that we can naturally identify S 0 (N), S 1 (N), and S(N) with the Riemann surfaces Y 0 (N)) C, Y 1 (N) C, and Y (N) C, respectively. First note that every member of S 0 (N) is represented by an enhanced elliptic curve of the form (E τ, 1/N + Λ t ) where τ h, Λ τ = Z + τz, and E τ = C/Λ τ. Furthermore we have (E τ, 1/N + Λ τ ) (E τ, 1/N + Λ τ ) precisely when τ = γτ for some γ Γ 0 (N), so S 0 (N) bijects naturally with Y 0 (N) C. And similarly we can identify S 1 (N) with Y 1 (N), and S(N) with Y (N). Now T acts on the group of divisors for S 1 (N) by T p [E, Q] = C [E/C, C + Q], 16

17 where C varies over the order p subgroups of E, and d [E, Q] = [E, dq]. We can also use this action to define an action of T on the Picard group of X 1 (N) C. We begin by defining an action of T on Div(X 1 (N)). We have T p : X 1 (N) Div(X 1 (N)) defined by, Γ 1 (N)τ Γ 1 (N)β, β where β ranges over a set of representatives of, ( ) 1 0 Γ 1 (N)\Γ 1 (N) Γ 0 p 1 (N), and d : X 1 (N) X 1 (N) defined by, Γ 1 (N)τ Γ 1 (N)ατ, where α is a member of Γ 0 (N) with bottom-right entry congruent to d modulo N. Both of these maps naturally extend to maps Div(X 1 (N)) Div(X 1 (N)), in a way that commutes with the inclusion S 1 (N) Y 1 (N) X 1 (N) given by [E τ, 1/N + Λ τ ] Γ 1 (N)τ. A less obvious but nonetheless true fact is that this action can be restricted to the Picard group of X 1 (N), so that the diagrams Div 0 (S 1 (N)) T p Div 0 (S 1 (N)) Pic 0 (X 1 (N)) T p Pic 0 (X 1 (N)) and Div 0 (S 1 (N)) d Div 0 (S 1 (N)) Pic 0 (X 1 (N)) d Pic 0 (X 1 (N)) commute. Finally because Pic 0 (X 1 (N) C ) is isomorphic to the Jacobian J 1 (N) we have an action of T on J 1 (N). To f we associate an ideal I f of T defined by I f = {T T : T f = 0}, and an abelian variety A f = J 1 (N)/I f J 1 (N). 17

18 Now we have an isomorphism given by T O f, where O f = Z[a n (f) : n 1], T p a p (f) and d ε(d) We then find that the fraction field of O f is the number field L attached to f, and using the above isomorphism we obtain an action of O f on A f. Moreover if d = [L : Q], then A f is a torus of complex-dimension d, so the l-adic Tate module of A f is isomorphic to Z 2d l. The action of T on A f defines an action of O f on T l (A f ). For each prime power l n we have a natural surjective map of l n -torsion groups Pic 0 (X 1 (N))[l n ] A f [l n ], which is surjective with kernel stable under G Q. So G Q acts on T l (A f ) also. And this action commutes with that of T. Putting all this together gives a Galois representation ρ Af,l : G Q GL 2d (Ẑ). Because T l (A f ) is an O f -module, the tensor product T l (A f ) Z Q is an L Q Q l - module, and moreover it is free of rank 2, see [3, lemma 9.5.3], so we have a representation ρ f,l : G Q GL 2 (L Q Q l ). An important result in the development of representations of modular forms is the Eichler-Shimura relation. For this we need a way to reduce the modular curve X 1 (N) modulo p, where p N is a prime number. To do this we use the moduli interpretation Y 1 (N) C = S1 (N). This gives us that Y 1 (N) is a scheme over Z[1/N]. To be more accurate, let R be a Z[1/N]-algebra, and F (R) be the set of isomorphism classes of pairs (E, P ) where E is an elliptic curve over R and P E(R) is a point of order N. This functor is representable by a Z[1/N]-scheme Y 1 (N) (here we do not mean the Q-rational points inside Y 1 (N) C ). Extending our scheme to include the cusps of Y 1 (N) we obtain the scheme X 1 (N) over Z[1/N], which is an integral model of X 1 (N) C. We then write X 1 (N) = X 1 (N) Spec Z[1/N] Spec F p. Theorem (Eichler-Shimura relation). Let p N. The Hecke operator T p acts on Pic 0 ( X 1 (N)) by T p = σ p + σ t p, where σ p denotes the Frobenius map in G Fp and σ t p is the dual of σ p. Proof. For a proof of this theorem see chapter 8 of [3]; the theorem itself is given at the end of the chapter as theorem This relation leads to the following corollary describing the action of the Frobenius elements. 18

19 Corollary Let p ln, then the action of Frob p on (L Q Q l ) 2 satisfies Frob 2 p T p Frob p +p = 0. Proof. By [3, theorem 9.5.1], the action of Frob p on T l (Pic(X 1 (N)) satisfies the equation above. Hence it satisfies that same equation on A f, and hence also on (L Q Q l ). Finally we are able to describe the Galois representation attached to f in the following theorem. Theorem (Eichler, Shimura). Let f S 2 (N, χ) new be a normalised eigenform with number field L, and let l be a prime number, then there exists a Galois representation ρ f,l : G Q GL 2 (L Q Q l ), such that if p is a prime number not dividing ln, then ρ f,l is unramified at p, and ρ f,l (Frob p ) is a root of x 2 a p (f)x + χ(p)p = 0. Moreover, ρ f,l is unique up to conjugation. Proof. See [3, theorem 9.5.4]. Constructing the representations of eigenforms with weight greater than 2 is substantially more complicated, and involves the use of étale cohomology to construct a space upon which G Q acts in an appropriate way. Therefore we will simply state the result, which is very similar to the weight 2 case. Theorem (Deligne [2]). Let k 2, f S k (N, χ) new be a normalised eigenform with number field L, and let l be a prime number, then there exists a Galois representation ρ f,l : G Q GL 2 (L Q Q l ), such that if p is a prime number not dividing ln, then ρ f,l is unramified at p, and ρ f,l (Frob p ) is a root of x 2 a p (f)x + χ(p)p k 1 = 0. Moreover, ρ f,l is unique up to conjugation. 19

20 2 Why study Galois representations of modular forms and elliptic curves? One of the major insights of the 20th century was the Taniyama Shimura Weil conjecture, which stated in loose terms that the representations of elliptic curves over Q are also representations of eigenforms of weight 2. This conjecture started receiving a lot of attention in the eighties when Frey and Ribet demonstrated that if proven true, the conjecture would prove Fermat s last theorem, and it was by proving that the conjecture holds for a certain class of elliptic curves that Wiles was able to prove Fermat s last theorem. In this chapter we will briefly explain why Wiles work was sufficient to prove Fermat s last theorem. 2.1 The modularity theorem Let E be an elliptic curve over Q, then we may assume without loss of generality that E is given by E : y 2 z + a 1 xyz + a 3 yz 2 = x 3 + a 2 x 2 z + a 4 xz 2 + a 6 z 3, a 1,..., a 6 Z. This equation is called a Weierstrass equation, and has the advantage of being welldefined over all commutative rings, and so in particular we may consider the variety Ẽ over F p given by reducing the equation of E modulo p (in this section p and l will always denote prime numbers). We say that E has good reduction at p if Ẽ does not have any singularities, and that E has bad reduction at p otherwise, in which case the singularity of Ẽ lies in Ẽ(F p) and is unique. The only primes where E may have bad reduction are those primes dividing the discriminant of E, so we may associate a positive integer N to E which encodes the type of reduction E has at each prime number, that is to say p N if and only if E has bad reduction at p, and if p N then the exponent of p in N encodes the type of singularity E(F p ) has. We call N the conductor of E. In the fifties and sixties Yutaka Taniyama, Goro Shimura, and André Weil came up with the Taniyama Shimura Weil conjecture which has since been proven and is now called the modularity theorem which suggested that all elliptic curves over Q are modular. There exist many equivalent definitions of what it means for an elliptic curve to be modular, but the definition most relevant to this thesis states that E is modular if there exists f S 2 (Γ 0 (N)), a finite prime λ of L = Q(a n (f) : n 1), and a prime number l such that Q l = Lλ, and their l-adic and λ-adic representations are isomorphic over Q l, i.e. ρ E,l ρ f,λ : G Q GL 2 (Q l ). 20

21 In this case we also say that E is modular by f with respect to l. To see why this is nice we define a new quantity: It then turns out that a p (E) = p + 1 #E(F p ). Tr ρ E,l (Frob p ) = a p (E), and so the modularity theorem tells us that there exists f S 2 (Γ 0 (N)) satisfying a p (E) = a p (f) whenever p ln to see this compare the characters of ρ f,λ and ρ E,l when E is modular by f with respect to l. This version of the modularity theorem further implies that a p (E) = a p (f) for all p; instead of proving this we direct the reader to page 392 of [3]. Now, because the trace is a continuous map from GL 2 (Q l ) to Q l, and because the Frobenius elements Frob p with p ln are dense (recall Chebotarev s theorem), we can now go a step further and say that E is modular by some f with respect to all l. To conclude this section we briefly give an example of a use for this theorem. Consider the elliptic curve E : y 2 z + yz 2 = x 3 x 2 z. This curve has conductor 11, and S 2 (Γ 0 (N)) = Cf, where f is the eigenform f = q n 1(1 q n ) 2 (1 q 11n ) 2. So according to the modularity theorem we have that a p (E) = a p (f) for all p. Also, the elliptic curves E : y 2 z + y = x 3 x 2 z 10xz 2 20z 3 and E : y 2 z + yz 2 = x 3 x 2 z 7820xz z 3 also have conductor 11, and so because f is the only candidate eigenform for elliptic curves of conductor 11 to be modular by, we find that for all primes p. 2.2 Fermat s last theorem a p (E) = a p (E ) = a p (E ) The Taniyama Shimura Weil conjecture received a lot of attention in the eighties when it was shown that a counter-example to Fermat s last theorem would provide a counterexample to the conjecture. In the nineties Wiles was able to show Theorem (Fermat s last theorem). Let l 3 be an integer, then there do not exist integers a, b, c 0 with a l + b l = c l. 21

22 We can immediately see that it suffices to consider the cases where l is a prime number, gcd(a, b, c) = 1, b is even, and 4 a+1. Let us suppose that a, b, c, l is a counter-example to Fermat s last theorem. In 1984 Gerhard Frey noted that such a counter-example would give rise to an elliptic curve E defined by E : y 2 z = x(x a l z)(x + b l z). The curve E is called either a Frey curve or a Frey Hellegouarch elliptic curve. Assuming the Taniyama-Shimura-Weil conjecture to be true, E would then be modular. Frey believed that in fact E could not be modular, and so a counter-example to Fermat s last theorem would disprove the Taniyama-Shimura-Weil conjecture. In 1987 [19] Serre then came up with the epsilon conjecture, and demonstrated that this conjecture would be sufficient to prove Frey correct, and in 1990 [12] Ribet was able to prove this conjecture, and so this conjecture is now called Ribet s theorem. The upshot was that there would exist f S 2 (Γ 0 (2)) and a finite prime λ of L λ = Q l, but instead of ρ f,λ ρ E,l we would have ρ E,l ρ f,λ : G Q GL 2 (F l ), where ρ E,l and ρ f,λ are the reductions of ρ E,l and ρ f,λ modulo l and λ respectively. However such an f cannot exist because the genus of X 0 (2) is 0, giving S 2 (Γ 0 (2)) = {0}. Thus proving Fermat s last theorem was reduced to proving that the Taniyama-Shimura- Weil conjecture holds for Frey curves, which Andrew Wiles and his student Richard Taylor were able to do in [23] and [22], when they proved that the conjecture holds for semi-stable elliptic curves elliptic curves E over Q such that if E has bad reduction at p then E(F p ) has a double point which includes the Frey curves. The full conjecture was later proven by Breuil, Conrad, Diamond, and Taylor, and so is now called the modularity theorem. 22

23 3 Images of adelic representations of modular forms The purpose of this chapter is to explain and prove the main results of the sections Large image results for one modular form, wherein Loeffler tries to find an analogue for Serre s theorem that the representations of elliptic curves have open image, and Joint large image in Loeffler s paper [8], where Loeffler discusses images of Cartesian products of Galois representations. First we pick an eigenform f, construct an open subgroup H of G Q, and reduce the codomain of the representations ρ f of f such that the restriction ρ f H can be said to be open. Then we find that if g is a second eigenform with coefficients independent to those of f then the image of ρ f ρ g H can still be said to be open in our new codomain. Remark We should mention that Loeffler s original motivation for his paper is his fourth section, which discusses the existence of certain elements in the image of representations of the form ρ f,p ρ g,p. In this thesis we will be focusing on the two preceding sections. 3.1 Is the image open? Recall theorem 1.2.1, which stated that if E is a elliptic curve without complex multiplication, then the image of ρ E in GL 2 (Ẑ) is open. On a purely intuitive level we might expect this result to generalise to modular forms by simply saying that Galois representations of modular forms also have open image in GL 2 (A Q ). However, we will now see that this obvious generalisation does not work. As before let f be a normalised cuspidal newform of weight k 2, level N, nebentypus ε, q-expansion n=1 a nq n, Fourier coefficient field L = Q(a n : n 1), and with Galois representation ρ f : G Q GL 2 (L Q A Q ). Let p be a prime of L lying over the prime number p, and let l be a prime number not dividing pn. By theorem 1.4.4, we see that det ρ f,p (Frob l ) = ε(l)l k 1, so there exists some finite power of det ρ f,p that takes values in Z p, which typically does not have finite index in O L,p, in particular whenever Q p is a proper subfield of L p. Over the course of the next few sections we will formulate an analogue to Serre s result for elliptic curves which will apply to modular forms without complex multiplication of weight k 2. 23

24 3.2 Inner Twists Let I f denote the set of field inclusions L C. Because L is a number field it is natural to ask, what happens if we apply some σ I f to the q-coefficients of f? The answer is that we get another modular form, f σ = a σ nq n, n=1 of the same weight and level. We call f σ a conjugate of f. Alternatively let χ be a primitive Dirichlet character with conductor r, then there exists a unique newform f χ with the property a p (f χ) = χ(p)a p (f) for all but finitely many prime numbers p (see section 3 of [15]). The newform f χ has the same weight as f, but if f χ has level M then we can only say that Nr and Mr have the same prime factors. If it happens that f σ = f χ then (σ, χ) is called an inner twist of f. If there exists a non-trivial Dirichlet character χ such that f χ = f, then f is said to have complex multiplication. From here on we will assume that f does not have complex multiplication. This means that given an inner twist (σ, χ), the character χ is uniquely determined by σ. Therefore, letting Γ be the set of inner-twists of f, we have an inclusion and we write χ = χ σ. Γ I f, Remark Saying that f has complex multiplication is equivalent to saying that there exists an imaginary quadratic field K such that a p = 0 whenever p is inert in K. In this case the imaginary quadratic field K is unique, and as with elliptic curves with complex multiplication we have an intrinsic link between modular forms with complex multiplication and imaginary quadratic fields. Next we have an important result about the set of inner-twists of f: Lemma (Momose [10, proposition 1.7]). If (σ, χ) is an inner-twist of f then σ(l) = L, and so Γ is a subset of Aut(L/Q). Moreover, if (γ, χ), (σ, µ) are inner twists of f, then (γ σ, χ σ µ) is also an inner twist, and so Γ is a subgroup of Aut(L/Q). Furthermore Γ is abelian. Finally, writing F for the subfield of L fixed by Γ, the field extension L/F is a Galois extension, with Galois group Γ. Proof. Here we write out the proof given by Momose. Let ρ f,l be the representation given by ρ f,l : G Q ρ f,l GL 2 (L Q Q l ) GL 2 (L Q Q l ). 24

25 Now note that L may be viewed as a subfield of Q l, and Q l can be viewed as a subfield of C, and moreover each embedding σ I f has image inside Q l. Therefore for each σ I f we let σ be the composite σ : L Q Q l Q l x y σ(x)y. For each σ I f, let ρ σ f,l be the composite ρ σ f,l : G Q ρ f,l GL 2 (L Q Q l ) σ GL 2 (Q l ). Let H be an open subgroup of G Q. By [14, 4.4] we know that ρ σ f,l H is irreducible for each σ I f. Moreover if H is normal, then ρ σ f,l H and ρ τ f,l H are not equivalent for all σ τ I f, for otherwise there would exist a Dirichlet character χ 1 such that ρ σ f,l is equivalent with ρ σ f,l χ, making f a newform with complex multiplication. Now, let (σ, χ) be an inner twist of f, so that ρ σ f,l and ρ f,l χ are equivalent representations. This gives det ρ σ f,l = χ2 det ρ f,l, or equivalently χ 2 = ε σ /ε, where χ and ε are viewed as characters on G Q in the usual manner. When ε has even order we have ε σ = ε 2j+1 for some integer j, giving χ 2 = ε 2j, and otherwise we have ε σ = ε 2j, so that χ 2 = ε 2j 1 = ε 2j for some integer j. In either case we have χ 2 = ε 2j for some integer j, thus χ = λ ε j with λ a character of order 1 or 2, and ε σ = ε 1+2i. This proves that χ has image inside L, and hence so does σ, therefore given two inner-twists (σ, λε i ) and (τ, µε j ), we have a στ p = (µ ε j )(p) σ a σ p = (λ µ ε i+j+2ij )(p)a p = a τσ p, for all but finitely many prime numbers p. The statement about composing inner-twists is now clear, thus Γ is an abelian subgroup of Aut(L/Q). To show that L/F is an abelian extension it suffices to show that Aut(L/F ) = Γ, so let σ Aut(L/Q) be such that σ F = 1. It follows that ρ σ f,l Gal(Q/F ) = ρ f,l Gal(Q/F ) and so there exists a Dirichlet character χ such that (σ, χ) is an inner-twist of f, hence σ Γ. For n > 0 let µ n denote the group of n-th roots of unity, and let µ denote the group of all roots of unity. For each n there exists a group isomorphism Gal(Q(µ n )/Q) = (Z/nZ). It follows that we may consider a Dirichlet character χ to be a character G Q Gal(Q(µ n )/Q) (Z/nZ) χ C, 25

26 where n is the conductor of χ. Finally we take H to be H = σ Γ ker χ σ G Q. Because H is the intersection of kernels of Dirichlet characters we can see that H is open in G Q, and because (complex conjugation, ε 1 ) is an inner-twist of f, the determinant of ρ f,l H is simply χ k 1, where χ is the cyclotomic character. 3.3 The quaternion algebra B For the reader s convenience let us restate the notation we are using. We take f = n 1 a nq n to be a normalised cuspidal newform of weight k 2 on Γ 1 (N) with nebentypus ε. We assume that f does not have complex multiplication. We write L = Q(a n : n 1) for the Fourier coefficient field of f, and we let Γ Aut(L/Q) be the group of inner twists of f. Each inner twists gives a Dirichlet character G Q C, and we denote the intersection of the kernels of these characters by H. We recall also that L is an abelian extension of F with Galois group Γ. Also, if (γ, χ) Γ then we write χ = χ γ, and we often refer to (γ, χ γ ) as just γ. In the first subsection we will describe a quaternion algebra B over F such that ρ f,l (H) has open image inside (B Q Q l ) for all prime numbers l, and full image for all but finitely many l. The remainder of this chapter is then devoted to showing that ρ f H has open image inside (B Q A Q ). We will then have an analogue of Serre s result that images of adelic representations of elliptic curves without complex multiplication are open. Let B be a quaternion algebra over F. For ease of notation we will now write G l = {x (B Q R) : norm B/F x Z (k 1) l } and G = {x (B Q A Q ) : norm B/F x Ẑ (k 1) }. (Here by Z (k 1) l and Ẑ (k 1), we mean the groups of (k 1)-th powers.) Later we will be introducing algebraic groups G and G over Q and Z respectively, so the use of G l and G will only be temporary. Theorem [Momose, Ribet] There exists a quaternion algebra B over F with an embedding B M 2 (L) such that ρ f (H) (B Q A Q ) GL 2 (L Q A Q ). Furthermore, the image ρ f,l (H) is open in G l for all prime numbers l, and for all but finitely many prime numbers l we have B Q Q l = M 2 (F Q Q l ), and rho f,l (H) = M 2 (O F Z Z l ) G l. 26

27 3.3.1 Constructing B The construction of B is originally due to Momose in his paper [10], wherein he studies the images of l-adic representations of modular forms. He begins by defining a Q-vector space V that has a linear L-action and satisfies dim Q L = 2. In particular, for each prime number l, the representation ρ f,l gives V l = V Q Q l the structure of a G Q - module. For each γ Γ, he constructs a Q-automorphism η γ of V that extends to an H-endomorphism of V l, in particular, if p is a prime number not dividing ln, then theorem 2.16 of Momose s paper says that ρ f,l (Frob p )η γ = η γ ρ f,l (Frob p )χ γ (p). The construction of η γ is quite complicated, and we have not developed the necessary terminology in this thesis, so the curious reader is directed to section 2 of Momose s paper. Finally, he gives the following theorem. Theorem (Momose, [10, 3.1]). The vector space X = γ Γ L η γ is a simple central F -algebra with maximal subfield L. Moreover, it can be described as follows: for each primitive Dirichlet character of conductor r, let g(χ) = r 1 i=0 χ(u)e2πiu/r, and define c : Γ Γ L by c(γ, σ) = g(χ 1 γ χ 1 σ )/g(χ 1 γσ ). Then c is a normalised 2-cocycle in H 2 (Γ, L ). Then, if α L and γ, σ Γ, we have η γ η σ = c(γ, σ)η γσ and η γ α = α γ η γ. So, we have an inclusion X End Q V. Also the centraliser B of X inside End Q V is a quaternion F -algebra. In Ribet s paper [15], which appeared at the same time as Momose s paper but was written independently, he considers the case where f has weight 2, and so has an abelian variety A attached to it. In section 5, he examines the algebra X = (End Q A) ZQ, and determines that it is a simple central F -algebra with maximal subfield L. In particular, it is also described using the same c H 2 (Γ, L ) as Momose uses. Furthermore, by showing that c has order at most 2 in H 2 (Γ, L ) he demonstrates that X has order 1 or 2 in the Brauer group Br(L/F ) of simple central F -algebras with maximal subfield L. This means that X = M #Γ (F ) or X = M #Γ/2 (D), where D is a division quaternion F -algebra. Later, in [16], he considers the action of X upon V constructed by Momose, and demonstrates that if X = M #Γ (F ) then B = M 2 (F ), and otherwise B = D. So if we know what Γ is, we have a concrete way to go about giving an explicit description of B, along with its splitting behaviour Properties of B In this section we give some important properties of B, all of which are due to either Momose or Ribet. Momose demonstrated that for each prime number l, the restriction of ρ l to H gives a map ρ l H : H (B Q Q l ). From the way in which H is defined, the determinant of ρ l H is simply χ k 1 l, where χ l is the l-adic cyclotomic character. Let H l be the image of H under ρ l. Due to Momose, we then have the following theorem: 27