Hecke Operators. Hao (Billy) Lee. November 2, 2015

Transcription

1 Hecke Oerators Hao (Billy) Lee November 2, 2015

2 Chater 1 Preliminary Everything in the reliminary section follows the notations and denitions from Modular Forms Denition. For SL 2 (Z), τ = aτ+b cτ+d for all τ H where H is the comlex half lane. We can ( extend action to the grou GL + 2 (Q) to act on Q { } by m ) n = am+bn cm+dn. Denition. For N N, dene the rincial congruence subgrou of level N to be Γ(N) = { c d SL 2 (Z) : c d mod N and say a subgrou of Γ of SL 2 (Z) is a congruence subgrou if Γ(N) Γ for some N N. Γ 0 (N) = { c d SL 2 (Z) : c d 0 mod N { } 1 Γ 1 (N) = SL 2 (Z) : mod N 0 1 Note that by taking Γ 0 (N) (Z/NZ) by d mod N is a surjective homomorhism with kernel Γ 1 (N). This shows that Γ 1 (N) is normal in Γ 0 (N), and the quotient is isomorhic to (Z/nZ). Denition. For any γ = GL + 2 (Q), dene the factor of automorhy j(γ, τ) C for τ H to be j(γ, τ) = cτ + d. For such a γ, we can dene the weight k oerator γ k on functions f : H C by } } (f γ k ) (τ) = (det γ) k 1 j(γ, τ) k f (γ(τ)) for τ H. For a congruence subgrou Γ of SL 2 (Z), we say that a meromorhic function f : H C is weakly modular of weight k with resect to Γ, if f γ k = f for all γ Γ. That is, f (γ(τ)) = j (γ, τ) k f(τ). 1

3 CHAPTER 1. PRELIMINARY 2 f is a modular form of weight k with resect to Γ, if it is holomorhic, weight-k invariant under Γ and f α k is holomorhic at for all α SL 2 (Z). If in addition, the rst coecient of the Fourier exansion of f α k is zero for all α SL 2 (Z), then f is a cus form. We denote the set of modular forms of weight k with resect to Γ by M k (Γ), and cus forms by S k (Γ). 1.2 Modular Curves Denition. Let Γ SL 2 (Z) be a congruence subgrou. Dene the modular curve Y (Γ) = Γ\H = {Γτ : τ H} to be the sace of orbits of Γ acting on H. In articular, denote Y 0 (N) = Γ 0 (N)\H, Y 1 (N) = Γ 1 (N)\H and Y (N) = Γ(N)\H. We should note here that, technically, Y (Γ) is a curve, which is the set of solutions of some given equation. what we are really dening here is Y (Γ)(C). Denition. The set of enhanced ellitic curve for Γ 0 (N), denoted S 0 (N), consists of ordered airs (E, C) where E is an ellitic curve and C is a cyclic subgrou of E of order N. (E, C) (E, C ) if there is an isomorhism of E and E taking C to C. The set of enhanced ellitic curve for Γ 1 (N), denoted S 1 (N), consists of ordered airs (E, Q) where E is an ellitic curve and Q is a oint of order N. (E, Q) (E, Q ) if there is an isomorhism of E and E taking Q to Q. The set of enhanced ellitic curve for Γ(N), denoted S(N), consists of ordered airs (E, (P, Q)) where E is an ellitic curve and (P, Q) are oints in E that generates E N with Weil airing e N (P, Q) = e 2πi/N. (Recall that E N = (Z/NZ) 2 ). (E, (P, Q)) (E, (P, Q )) if there is an isomorhism of E and E taking P to P and Q to Q. Theorem. 3, Thm Modulo details, there are bijections S 0 (N) = Y 0 (N), S 1 (N) = Y 1 (N) and S(N) = Y (N). Examle. For N = 1, Y 0 (1) = Y 1 (1) = Y (1) = SL 2 (Z)\H. Recall that an ellitic curve can be determined by a lattice generated by 1 and some τ H. Two lattices generated the same ellitic curve if τ SL 2 (Z)τ. This agrees with our theorem. Y (Γ) can be made into a Riemann surface (1 dimension comlex manifold) by taking the quotient toology obtained from the quotient ma π : H Γ by τ Γτ. We can comactify Y (Γ) to get X(Γ) = SL 2 (Z)\ (H Q { }). The extra oints are called the cuss. X(Γ) is Hausdorf, connected and comact 3, Pro If f is weight k invariant with resect to Γ, then f is a degree k homogenous function on modular curves with resect to Γ. For details, see 3, Pg 41.

4 Chater 2 Hecke Oerators We will motivate Hecke Oerators following 3 by introducing double coset oerators. 2.1 Double Coset Denition. Let Γ 1 and Γ 2 be congruence subgrous and let α GL + 2 (Q), dene Γ 1 αγ 2 = {γ 1 αγ 2 : γ 1 Γ 1, γ 2 Γ 2 } to be the double coset in GL + 2 (Q). The grou Γ 1 acts on Γ 1 αγ 2 by left multilication, artitioning it into orbits. It can be shown that the number of orbits is nite 3, g 164. Suose Γ 1 αγ 2 = j Γ 1 β j where {β j } are the orbit reresentatives Denition. 3, Def For congruence subgrous Γ 1 and Γ 2 of SL 2 (Z) and α GL + 2 (Q), the weight-k Γ 1αΓ 2 k oerator takes functions f M k (Γ 1 ) to f Γ 1 αγ 2 k = j fβ j k This is well-dened 3, Exercise In fact, we have the following theorem. Theorem. Γ 1 αγ 2 k : M k (Γ 1 ) M k (Γ 2 ) and S k (Γ 1 ) S k (Γ 2 ). Proof. The full roof can by found on age 165 of 3. Here, we will only show invariance. For all γ Γ 2, the ma Γ 1 \Γ 1 αγ 2 Γ 1 \Γ 1 αγ 2 given by Γ 1 β Γ 1 βγ 2 is well-dened and bijective. Therefore, (f Γ 1 αγ 2 k ) γ k = j f β j γ k = f Γ 1 αγ 2 k. Secial cases 3: 1. When Γ 1 Γ 2, with α = I then Γ 1 αγ 2 k is the natural inclusion of M k (Γ 1 ) into M k (Γ 2 ). 2. Γ 1 Γ 2. Taking α = I again, and letting {γ 2,j } be the set of coset reresentatives for Γ 1 \Γ 2 makes the double coset oerator f Γ 1 αγ 2 k = j f γ 2,j k the natural trace ma that rojects M k (Γ 1 ) onto M k (Γ 2 ) by symmetrizing over the quotient. 3

5 CHAPTER 2. HECKE OPERATORS 4 3. If α 1 Γ 1 α = Γ 2 then f Γ 1 αγ 2 k = f α k, the natural translation, is an isomorhism. 2.2 T n and d Denition. Let Γ 1 = Γ 2 = Γ 1 (N) and let α Γ 0 (N). Recall that Γ 0 (N)/Γ 1 (N) = (Z/NZ) by the ma d mod N. This shows that Γ 1 (N) Γ 0 (N), and we have f Γ 1 (N)αΓ 1 (N) k = f α k for all α Γ 0 (N) and f M k (Γ 1 (N)). This is case 3 from above. Note that this induces an action of α Γ 0 (N) on M k (Γ 1 (N)). Because Γ 1 (N) acts trivially on f, this really is an action of (Z/NZ) on M k (Γ 1 (N)). For d (Z/NZ), we can dene the Diamond Oerator d : M k (Γ 1 (N)) M k (Γ 1 (N)) by d f = f α k for any α = Γ 0 (N) with δ d mod N. c δ 1 0 Denition. Again, let Γ 1 = Γ 2 = Γ 1 (N). Let α = for some rime. Then dene 0 T P : M k (Γ 1 (N)) M k (Γ 1 (N)) by T f = f Γ 1 (N)αΓ 1 (N) k. Now, we will show that T and d commutes. For full detail, see age 169 of 3. To do this, rst observe that { } Γ 1 (N) Γ 1 (N) = γ M 2 (Z) : γ mod N, det γ = In fact, for any γ Γ 0 (N), γαγ 1 γ Γ 0 (N). Then 0 mod N. Suose that Γ 1 (N)αΓ 1 (N) = j Γ 1 (N)β j, and x Γ 1 (N)αΓ 1 (N) = Γ 1 (N)γαγ 1 Γ 1 (N) = γγ 1 (N)αΓ 1 (N)γ 1 by normality = γ j Γ 1 (N)β j γ 1 Hence, we have j Γ 1 (N)β j = γ j Γ 1 (N)β j γ 1 and thus j Γ 1 (N)γβ j = j Γ 1 (N)β j γ 1. Note, it need not be the same for each term. We can now show commutativity with this identity. Let γ Γ 0 (N) where the lower right corner entry is δ d mod N. Then d T f = d j f β j k = j f β j γ k = j f γβ j k = T d f for all f M k (Γ 1 (N)).

6 CHAPTER 2. HECKE OPERATORS 5 1 j m n 0 In fact, we can nd that β j = 3, Page for 0 j < and β = N 0 1 if N where m nn = 1 Proosition. 3, Pro T f = 1 j=0 f 1 j=0 f 1 j 0 1 j 0 k + f N k n k if N if N where m nn = 1 In other words, T f(τ) = ( ) j=0 f τ+j ( 1 j=0 f τ+j if N ) + k 1 f(τ) if N Note that in this last formula, it does not matter that f M k (Γ 1 ). In fact, with this algebraic formula, we can dene Hecke oerators on any congruence subgrou Γ. Now, we try to extend d and T P to all n Z +. For n Z + with gcd (n, N) = 1, dene n to be n mod N. If gcd (n, N) > 1, then dene n = 0. This denition makes multilicative on Z +. For rime owers r, dene T r = T T r 1 k 1 T r 2 for r 2. Then for n = ei i construction T n and d still commute. as its rime factorization, dene T n = T e i. By i 2.3 Modular Curve Interretation Let Γ 1 and Γ 2 be congruence subgrous of SL 2 (Z). Suose Γ 1 αγ 2 = j Γ 1 β j, where {β j } are coset reresentations. Let X 1 = X(Γ 1 ) and X 2 = X (Γ 2 ), then Γ 1 αγ 2 k : Div(X 2 ) Div(X 1 ) by Γ 2 τ j Γ 1β j (τ) 3, Pg 166. We will consider the case where the Hecke oerators act on Γ = Γ 1 (N) (don't care about the weight). We will now give a geometric interretation of this, following Remark 1.11 and section 1.3 of 4, and age 174 of 3. Recall that the modular curve Y 1 (N) is in bijective corresondence with S 1 (N). S 1 (N) consists of airs (E, Q) where E is an ellitic curve and Q is a oint of E of order N. For N, the moduli sace interretation is T P : Div (S 1 (N)) Div (S 1 (N)) by E, Q C E/C, Q + C where the sum is taken over all subgrous C of E of order such that C Q = {id E }. This comes from the fact that we have the following corresondence, Div (Y 1 (N)) Div (S 1 (N)) T T Div (Y1 (N)) Div (S1 (N)) Γ 1 (N)τ j Γ 1(N)β j (τ) Eτ, 1 N + Λ τ C Eτ /C, 1 N + C For more details about why this is true, see age 174 of 3. There is an isogeny from C/Λ to C/Λ if and only if there exists some m C such that mλ Λ. If N, then there are exact +1 distinct -isogenies from ( ) ( ) C/ τ, 1, 1 N. Their images are: C/ τ+j, 1, 1 N for j = 0,..., 1 and ( C/ τ, 1, ) N. If N, then we lose the last -isogeny, because the oint N is of order less than N. Note, these + 1 isogenies are exactly φ j (τ) = τ+j for j = 0,..., 1 and φ (τ) = τ. The ma f(τ) ω f = 2πif(τ)dτ is an isomorhism between S 2 (Γ) and Ω 1 (X Γ ) of holomorhiierentials on X Γ 4, ( Lemma ) ( ) This also ( shows ) that dim S 2 (Γ) is nite and equal to g = genus (X(Γ)). Notice that φ j (ω f ) = 2πif τ+j d τ+j = 2πi f τ+j dτ

7 CHAPTER 2. HECKE OPERATORS 6 for all j = 0,..., 1. Combining this fact, with the algebraienition of T, we see that for N, ω T(f) = φ j (ω f ). 2.4 Petersson Inner Product Denition. Dene the hyerbolic measure on the uer half lane dµ(τ) = dxdy y 2 for all τ H. We can extend the measure to H = H Q { } because Q { } has measure zero. This is invariant under under GL + 2 (R), so in articular, it's SL 2(Z)-invariant. Recall that D = {τ H : Re(τ) 12, τ 1 } { } is a fundamental domain of H under the action of SL 2 (C). It can be shown that for any continuous and bounded functions φ : H C and α SL 2 (Z), D φ (α(τ)) dµ(τ) converges. Let {α j } SL 2 (Z) be a set of coset reresentatives, so that SL 2 (Z) = j {±I} Γα j. Now, consider φ : H C in M k (Γ). Since φ and dµ are are Γ invariant, we have ˆ ˆ φ (α j (τ)) dµ(τ) = D φ(τ)dµ(τ). α jd (2.4.1) j Furthermore, α j D reresents X(Γ) u to some boundary identication, so we can dene equation (2.4.1). X(Γ) φ(τ)dµ(τ) to be Denition. For a congruence subgrou Γ, dene the volume of Γ to be V Γ = dµ(τ). X(Γ) Fact. V Γ = SL 2 (Z) : {±} Γ V SL2(Z). Denition. Let Γ SL 2 (Z) be a congruence subgrou. Dene the Petersson Inner roduct by, Γ : S k (Γ) S k (Γ) C f, g Γ = 1 ˆ f(τ)g(τ) (Im(τ)) k dµ(τ) V Γ X(Γ) It can be shown that this is well-dened (f(τ)g(τ) (Im(τ)) k is Γ invariant and the integral converges). Additionally, it is not hard to see that this is linear in the rst variable, and conjugate linear in the second. Additionally, it's Hermitian-symmetric and ositive denite. The reason for 1 V Γ is so that if Γ Γ, then, Γ =, Γ on S k (Γ). This is only dened on cus forms because because the inner roduct does not converge on all of M k (Γ). For Γ SL 2 (Z) a congruence subgrou and α GL + 2 (Q), dene α = det(α)α 1. By comutation, we have that α k = α k and ΓαΓ k = Γα Γ k are their adjoints under the Petersson Inner Product 3, Pro In articular, on S k (Γ 1 (N)), and for N, we have adjoints: = 1 and T = 1 T. For this, we can show that n and T n for gcd (n, N) = 1, are all normal. By the Sectral Theorem of linear algebra, since S k (Γ 1 (N)) is nite dimensional, and n, T n for gcd (n, N) = 1 are a commuting family of normal oerators, there exists an orthogonal basis of simultaneous eigenvectors for the oerators. Let T denote the C-algebra generated by the all Hecke oerators T n and d. A modular form is an eigenform if it is a simultaneous eigenvector for all T T. Note that this does not form asis, because T is not semi-simle. Let T 0 denote the set of all T n and n where gcd(n, N) = 1. orthogonal basis of simultaneous eigenforms. This algebra is semi-simle and so we have an

8 CHAPTER 2. HECKE OPERATORS Eigenforms Denition. If f S k (Γ) is an eigenform if it is a simultaneous eigenvector for all T T. If it has Fourier exansion f(τ) = n=1 a n(f)q n where a 1 (f) = 1 then we say f is normalized. Let f be an eigenform, then it has an associated algebra homomorhism λ f : T C where T f = λ(t )f for all T T. Additionally, we can dene χ : (Z/NZ) C by sending n to the eigenvalue of n corresonding to f, that is n f = χ(n)f. It can be shown that χ is a Dirichlet character. Proosition. 4, 1.17Given a non-zero algebra homomorhism λ : T C, there is exactly one eigenform, u to scaling, such that T f = λ(t )f for all T T. Proosition. 3, Pro Let f M k (N) with associated character χ. Then f is a normalized eigenform if and only if the coecients of the Fourier series satises the following: 1. a 1 (f) = 1 2. a r(f) = a (f)a r 1(f) χ() k 1 a r 2(f) for all rime and r 2 3. a mn(f) = a m (f)a n (f) when gcd (m, n) = 1 To summarize, a n (f) = a 1 (f)λ(t n ). Denition. For a modular form f M k (N) where χ is a Dirichlet character, dene its L-function to be L(s, f) = n=1 a nn s where f(τ) = n=0 a ne 2πinτ is its Fourier series exansion. With some work, the revious roosition shows that f is a normalized eigenform, if and only if its L function has an Euler roduct exansion 4, Thm , Thm L (s, f) = ( 1 a s + χ() k 1 2s) 1. Here, we take χ() = 0 for N.

9 Chater 3 Galois Reresentation The denitions and constructions in this chater come from various sections of Jacobian Recall that the ma f(τ) w f = 2πif(τ)dτ is an isomorhism between S 2 (Γ) and Ω 1 (X Γ ) of holomorhic dierentials on X Γ 4, Lemma This shows that dim S 2 (Γ) is equal to g = genus (X(Γ)). Let V = S 2 (Γ) v = Hom (S 2 (Γ), C) be the dual sace of S 2 (Γ), the weight 2 cus forms of some congruence subgrou Γ of SL 2 (Z). This is a comlex vector sace of dimension g = genus (X(Γ)). The integral homology Λ = H 1 (X(Γ), Z) mas naturally to V by sending a homology cycle c to the functional φ c where φ c (f) = c w f. The image of Λ is a discrete Z-module of rank 2g, so it can be viewed as a lattice in V. We call the comlex torus V/Λ, the Jacobian variety of X (Γ) over C. If Γ = Γ 0 (N) or Γ 1 (N), we will write J 0 (N) and J 1 (N) resectively. Fix τ 0 H. Dene the Abel-Jacobi ma Φ AJ : X(Γ)(C) J Γ by Φ AJ (P )(f) = τ 0 w f. This is well-dened and does not deend on the choice of ath. By linearity, we can extend this to a ma on Div (X(Γ)). Then we can restrict it down to the degree 0 divisors Div 0 (X(Γ)). Here, the Abel-Jacobi ma no longer deends on the base oint τ 0. Theorem. 4, Thm 1.15 (Abel-Jacobi Theorem). The ma Φ AJ : Div 0 (X(Γ)) J Γ has kernel consisting of recisely P (X(Γ)) which is the set of rincial divisors. Therefore, the ma induces an isomorhism between J Γ and the Picard grou, P ic 0 (X(Γ)) = Div 0 (X(Γ)) /P (X(Γ)). Hecke oerators act on V = (S 2 (Γ)) v via duality and they hold Λ stable. Hence, Hecke oerators give rise to endomorhisms of J Γ. Denition. A corresondence on a curve X is a divisor C on X X taken modulo {P } X and X {Q}. Let π 1 and π 2 denote the rojection of X X onto each of the factors. Then C induces a ma on Div(X) by C(D) = π 2 ( π 1 1 (D) C), where D 1 D 2 denotes intersection of the two divisors. C reserves the divisors of degree 0 and sends rincial divisors to rincial divisors. Hence C gives an algebraic endomorhism of Jac(X). We can in fact dene comosition of corresondences to get that the set of corresondences form a ring. See 6 for more details. Now, back to X (Γ). See age 32 of 4 for more details. We dene the Hecke corresondence T n to be the closure in X Γ X Γ of the locus of oints (A, B) in Y Γ Y Γ, where there is a degree n isogeny of ellitic curves with Γ structure from A to B. Let's examine a concrete examle, with Γ = Γ 1 (N) and let N. Consider the 8

10 CHAPTER 3. GALOIS REPRESENTATION 9 grah of T in (X 1 (N) X 1 (N)). This is a corresondence. Consider what the induced ma of T is on divisors. By denition, T ((E, P )) = π 2 ( π 1 1 ((E, P )) T ) = (E/C, P mod C) where the sum runs over the subgrous C of E with order. If (A, B) belongs to T then the isogeny dual to A B gives a -isogeny from B to A so that T v = 1 T. Let Γ = Γ 1 (N). Let φ X1(N) be the Frobenius morhism on X 1 (N) /F which is a degree isogeny that raises coordinates to the -th ower. Here, X 1 (N) /F is the reduction of the curve to characteristic. For more detail on how this is done, see age 36 of 4. Consider the grah of φ X1(N) in (X 1 (N) X 1 (N)) /F. It is a corresondence of degree, which will now be called F. Fix a oint (E, P ) X 1 (N) /F. Our goal is to comute T ((E, P )) using the Frobenius ma. Let (E, P ) = φ X1(N) ((E, P )). To nd the other ellitic curves -isogenous to E, we can consider the ellitic curves E, such that when we aly the Frobenius to it, we get E. To do so, we consider the transose corresondence F (interchange the two factors of X 1 (N) X 1 (N)). The corresonding endomorhism on J Γ induced by F is the dual endomorhism of φ JΓ. Consider the divisor F ((E, P )) = (E 1, P 1 ) (E, P ) Since φ Ei, the Frobenius endomorhism on E i, is an isogeny of degree from (E i, P i ) to (E, P ), we also have the dual isogeny from (E, P ) to (E i, P i ). If E is ordinary at then (E, P ), (E 1, P 1 ),..., (E, P ) are a comlete list of distinct curves with Γ-structure which are -isogenous to (E, P ). Hence, we have the following equality on divisors, T ((E, P )) = (E, P ) + (E 1, P 1 ) (E, P ) = (F + F ) ((E, P )). Since ordinary oints are dense in X 1 (N) /F, T = (F + F ) as endomorhisms of J 1 (N) /F. Theorem. 4, Thm 1.29 For N, the endomorhism of T of J Γ/Fq Eichler-Shimura congruence relation. satises T = F + F. This is called the 3.2 Shimura's Construction Denition. Let S 2 (Γ, Z) to be the sace of modular forms with integral Fourier coecients in S 2 (Γ). Given a ring A, dene S 2 (Γ, A) = S 2 (Γ, Z) A. Note, S 2 (T, C) = S 2 (Γ). Let T Z be the ring generated over Z by the Hecke oerators T n and d acting on S 2 (Γ, Z). Given a ring A, dene T A = T Z A. T A acts on S 2 (Γ, A) in a canonical way. Let f = n=1 a n(f)q n be an eigenform. Let K f be a number eld generated by all the a n (f)'s. Let λ f : T Q K f be associated algebra homomorhism. I f = ker λ f T Z. The image of I f (J Γ ) is a subabelian variety of J Γ which is stable under the actions of T Z and is dened over Q. Denition. Dene A f = J Γ /I f (J Γ ). It is an abelian variety dened over Q and deends only on f the orbit of f under G Q. Its endomorhism ring contains T Z /I f which is isomorhic to an order in K f. In fact, from the actions of T, we get an embedding K f End Q (A f ) Q 4, Pro A f is a comlex tori 4, Lemma Let V f of V = S 2 (Γ) v on which T acts on via λ f ({f : T f = λ(t )f T T}. V f has dimension 1 as a comlex vector sace4, Thm 1.22, Lemma Let π f be the orthogonal rojection of V onto V f relative to the Petersson inner roduct.

11 CHAPTER 3. GALOIS REPRESENTATION 10 Let f be all the eigenforms whose Fourier coecients are Galois conjugates to those of f. The number of forms is K f : Q. Let V f = g f V g and π f = g f π g which is simly the orthogonal rojection of V onto V f. It should be noted that π f T Kf and π f T Q. Lemma. 4, Lemma 1.46 The abelian variety is isomorhic over C to the comlex torus V f /π f (Λ) with the ma π f : V/Λ V f /π f (Λ) corresonding to the natural rojection from J Γ to A f. This also shows that A f is of dimension K f : Q. Proosition. 4, Proosition 1.53 The following are equivalent: The curve E is isogenous over Q to A f for some newform f on some congruence goru Γ There is a non-constant morhism dened over Q from X 0 (N) to E We won't discuss what newforms are, but basically we can decomose S k (Γ 1 (N)) into newforms and oldforms. For more information, see section 5.6 of 3. In articular, if E is an ellitic curve that satisfy the above roerty, then we say it is a modular ellitic curve. Conjecture. Shimura-Taniyama Conjecture 4, Conj All ellitic curves dened over Q are modular. Of course, we now know this is true for semi-stable ellitic curves (Andrew Wiles). Dene the Tate module of A f by T l (A f ) = lim (A f l n ). T l (A f ) Zl Q l is free K f Q l module of rank 2 4, Lemma Theorem. 4, Thm 1.41 For Nl, the characteristic olynomial of the Frobenius endomorhism F on T Ql -module T l (A f ) Q l is X 2 T X + = 0. Proof. By Eichler-Shimura relation. 3.3 Main Theorems For this section, we will let f = n=1 a n(f)q n be an eigenform of weight 2 and level N. Let χ : (Z/NZ) C be its associated character, such that d f = χ(d)f. Let K f be a number eld generated by all the a n (f)'s and values of χ. The action of the Hecke algebra on J 1 (N) rovides an embedding K f End Q (A f ) Q. Recall that T l (A f ) Zl Q l is a free K f Q l module of rank 2. The action of the Galois grou commutes with that of K f, so by choosing a basis for the Tate module, we get an interretation G Q GL 2 (K f Q l ). Because K f Q l can be identied with the roduct of comletions of K f at the rimes over l, we just induced an l-adic reresentation of G Q from f. Theorem. 5, Thm Suose k 2. Then for all rimes of K f, there exists an odd irreducible Galois reresentation ρ f, : G Q GL 2 ( (K f ) ) such that for all l rime to N and to, ρ f, is unramied at l, and the characteristic olynomial of ρ f, (F rob l ) is x 2 a l (f)x + χ(l)l k 1.

12 CHAPTER 3. GALOIS REPRESENTATION Suose k = 1. Then there exists an odd irreducible Galois reresentation ρ f : G Q GL 2 (C) such that for all l rime to N, ρ f x 2 a l (f)x + χ(l). is unramied at l, and the characteristic olynomial of ρ f (F rob l ) is Full roofs of these statements can be found in 1 and 2 for statements 1 and 2 resectively. The reason why the weight 1 case is stated in a searate statement is because it comes from Artin reresentations, and statement 1 comes from l-adic reresentations. For k = 2, J 1 (N) has good reduction at all rimes N. This shows that the action of the Galois grou on T l (A f ) Q l is unramied and is described by the Frobenius endormorhism φ on the Tate module of the reduciton. The characteristic olynomial of φ is X 2 T X + = 0 by the Eichler-Simura relation.

13 Bibliograhy 1 P. Deligne. Formes modulaires et rerésentations l-adiques. Séminaire Bourbaki, 11:139172, P. Deligne and J.-P. Serre. Formes modulaires de oids 1. Annales scientiques de l'école Normale Suérieure, 7(4):507530, F. Diamond and J. Shurman. A First Course in Modular Form. Information Security and Privacy, H. Darmon and F. Diamond and R. Taylor. Fermat's Last Theorem. Current Develoments in Mathematics, 1:1157, J. Weinstein. Recirocity laws and Galois reresentations: Recent Breakthrough A. Weil. Variétés abéliennes et courbes algébriques