Lecture 12 : Hecke Operators and Hecke theory

Transcription

1 Math 726: L-functions and modular forms Fall 2011 Lecture 12 : Hecke Operators and Hecke theory Instructor: Henri Darmon Notes written by: Celine Maistret Recall that we found a formula for multiplication of operators : T n T m = T nm when gcd(m, n) = 1 T p r = T p T p r 1 p k 1 T p r 2 In particular, we note that all Hecke Operator commute. Let T k be the algebra generated over C by the Hecke Operator (T 1, T 2,...) in End C (M k (SL 2 (Z))), ie T k M d (C) where d = dim C (M k (SL 2 (Z))) Definition 1. A modular form f M k (SL 2 (Z)) is called an eigenform or Hecke eigenform if it is a simultaneous eigenvector for all the Hecke Operators: T n (f) = λ n f ( λ n C withf 0) Also, the eigenform f is said to be normalized if a 1 (f) = 1 Remark 1. If f is a normalized eigenform we have : but T n (f) = λ n f for some λ n a n (f) = a 1 (T n (f)) = a 1 (λ n f) = λ n a 1 (f) = λ n since a 1 (f) = 1. Hence we can recover the Fourier coefficients of a normalized eigenform by considering the associated eigenvalues of Hecke Operators : λ n = a n (f). This leads to the following property : Multiplicity one property : An eigenspace for any collection {λ n } of eigenvalues for T n is exactly 1-dimensional, spanned by the normalized eigenform. 1

2 Theorem 2. (Hecke) If f is a normalized eigenform on SL 2 (Z) and if L(f, s) is its associated L-function then for Re(s) > k/2 + 1 L(f, s) = p (1 a p p s + p k 1 2s ) 1 proof : Relations among the T n s and T p r s imply that if f is a normalized eigenform : a(mn) = a(m)a(n) if (m, n) = 1 and thus a(p r ) = a(p)a(p r 1 ) p k 1 a(p r 2 ) L(f, s) = a n n s n=1 = ( a p rp rs) p r=0 = (1 a(p)p s + p k 1 2s ) 1 p We now see some examples of eigenforms and Dirichlet Series : Eisenstein Series : G k M k (SL 2 (Z)) is an eigenform. Normalizing appropriately, we get G k with a 1 ( G k ) = 1 Now recall that the Fourier expansion of G k involves δ k 1 (n) = d n dk 1, we have : a n ( G k ) = δ k 1 (n) = d n d k 1 where δ k 1 is multiplicative and satisfy : δ k 1 (p r ) = δ k 1 (p)δ k 1 (p r 1 ) p k 1 δ k 1 (p r 2 ) 2

3 A cuspidal example : S 12 (SL 2 (Z)) is also an eigenform. Writing the q-exansion of : (q) = τ(n)q n n=1 where τ(n) is the n th Fourier coefficient, also called the Ramanujan τ-function, is in particular multiplicative. = q n (1 q n ) 24 Then L(, s) = n τ(n)n s = p (1 τ(p)p s + p 11 2s ) 1 Now, we can ask how often do we find eigenforms if we look at S k (SL 2 (Z)) when k > 2 is even. Theorem 3. (Hecke) The space M k (SL 2 (Z)) has a basis consisting of normalized eigenforms M k (SL 2 (Z)) = M k (SL 2 (Z)) Tn=φ(Tn) φ Hom Alg (T k,c) where M k (SL 2 (Z)) Tn=φ(Tn) := {f M k (SL 2 (Z)) T n (f) = φ(t n )(f)} proof : S k (SL 2 (Z)) is naturally equipped with a hermitian inner product, the Petersson scalar product < f, g >= y k f(z)g(z) dzdz y 2 which is non-degenerate and positive. SL 2 (Z)/H 3

4 Key fact : The operator T n is self adjoint with respect to this pairing: < T n (f), g >=< f, T n (g) > By diagonalisibility of self adjoint operators we have : S k (SL 2 (Z)) = S k (SL 2 (Z)) Tn=φ(Tn) φ Hom Alg (T k,c) which is an orthogonal direct sum. But since we know that M k (SL 2 (Z)) is a direct sum of the line spanned by the Eisenstein Series E k with the space of cusp forms, we also have such a decomposition for M k (SL 2 (Z)): M k (SL 2 (Z)) = C.E k Sk (SL 2 (Z)) Note however that < f, g > does not converge for f, g M k (SL 2 (Z)), hence is it not a orthogonal direct product. Proposition 1. If f S k (SL 2 (Z)) is a normalized eigenform then the Fourier coefficients a n (f) are algebraic numbers. More precisely, Q(a 1 (f),..., a n (f),...) is a totally real extension of Q of degree at most the dimension of S k (SL 2 (Z)). proof : As we saw before, S k (SL 2 (Z)) has a basis consisting of modular forms with rational Fourier coefficients. Let (f 1,..., f d ) be such a basis. T n preserves the rational vector space Qf 1 Qf2... Qfd and therefore can be viewed as a d by d matrix with rational entries relative to this basis. But the eigenvalues of such a matrix are algebraic, and moreover, if we let T k (Q) be the Q-algebra generated by T 2, T 3, T 4,... acting on S k (SL 2 (Z)), then T k (Q) is isomorphic to a commutative subring of M d (Q). Therefore, any φ Hom(T k (Q), Q) has its image contained in an extension of Q of degree less or equal to d = dim S k (SL 2 (Z)) Note : Hom(T k (Q)) is totally real by self adjointness. Remark 2. The first interesting case, where dim(s k (SL 2 (Z))) > 1 is k = 24 As a possible exercise, one can compute the field of definition of the Fourier coefficients of eigenforms in S 24 (SL 2 (Z))) 4

5 This complete the study of Hecke Theory and L-functions of modular forms in case of level 1. It remains to study the theory when the level is arbitrary, and discuss the features that generalize and those that do not. 5