Converse theorems for modular L-functions

Transcription

1 Converse theorems for modular L-functions Giamila Zaghloul PhD Seminars Università degli studi di Genova Dipartimento di Matematica 10 novembre 2016 Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

2 Introduction modular form: f : H C holomorphic with suitable invariance properties; Dirichlet series: L(s) = a nn s Example (Riemann ζ function) ζ(s) = n s verifies Φ(s) = Φ(1 s) where Φ(s) = π s 2 Γ ( s 2 ) ζ(s). modular form Dirichlet series functional equation. QUESTION: is it true that a Dirichlet series satisfying a proper functional equation comes from a modular form? Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

3 Modular group and modular forms The (full) modular group is defined as { ( ) } a b SL 2 (Z) = γ = : a, b, c, d Z, ad bc = 1 c d ( ) a b γ = c d SL 2 (Z) is generated by the matrices ( ) 0 1 S = 1 0 γ : z az + b cz + d T = which correspond to the transformations on H ( ) S : z 1 z e T : z z + 1 Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

4 Modular group and modular forms Definition A modular form of weight k Z + for SL 2 (Z) is a function f holomorphic on H { } satisfying for any γ SL 2 (Z) ( ) az + b f(z) = f γ (z) = (cz + d) k f = (cz + d) k f(γz). cz + d f holomorphic at f(z) = Definition a n e 2πinz = a n e(nz). n=0 A cusp form is a modular form such that f( ) = a 0 = 0. n=0 Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

5 Modular group and modular forms Notation: Remark M k (SL 2 (Z)) is the space of modular forms of weight k for the full modular group. S k (SL 2 (Z)) is the space of cusp forms of weight k for the full modular group. 1 Since SL 2 (Z) = T, S a function f holomorphic on H is a modular form of weight k for SL 2 (Z) if and only if it satisfies f(z) = f(z + 1). f(z) = z k f( 1/z). 2 Since I SL 2 (Z), if k is odd then M k (SL 2 (Z)) = 0. In fact, f = f I f(z) = ( 1) k f(z). 3 The definition of the slash operator can be extended to γ GL + 2 (R) f γ (z) = (det(γ)) k/2 (cz + d) k f(γz). Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

6 The functional equation Given f S k (SL 2 (Z)), with sequence of Fourier coefficients (a n ), we associate the Dirichlet series L(s, f) = a n n s and the complete L-function Λ(s, f) = (2π) s Γ(s)L(s, f). The invariance property f = f S gives the modular relation f(z) = z k f( 1/z), which with standard analytic methods leads to the functional equation Λ(s, f) = i k Λ(k s, f). Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

7 Congruence subgroups Let N 1. The principal congruence subgroup of level N is { ( ) 1 0 Γ(N) = γ SL 2 (Z) : γ (mod N) 0 1 }. Definition A subgroup Γ SL 2 (Z) is a congruence subgroup if there exists an integer N 1 such that Γ(N) Γ. Γ 1 (N) = Γ 0 (N) = { γ SL 2 (Z) : γ { γ SL 2 (Z) : γ ( ) ( ) 0 Γ(N) Γ 1 (N) Γ 0 (N) } (mod N) } (mod N) Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

8 Modular forms for congruence subgroups Let Γ be a congruence subgroup.we define an equivalence relation over P 1 (Q) = Q { } z w there exists γ Γ such that w = γz. An equivalence class modulo Γ is a cusp. Definition A modular form of weight k for a congruence subgroup Γ is a function f holomorphic on H = H P 1 (Q), such that for all γ Γ f = f γ. A cusp form is a modular form vanishing at all cusps. Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

9 The functional equation ( ) 0 1 Observe that, if N > 1, the matrix S = / Γ (N). So, how can we get a functional equation ( in this case? 0 1 N 0 ). It normalises the group Consider the Fricke involution ω N = ( ) a b Γ 0 (N), i.e. if γ = Γ c d 0 (N), ( ) γ = ω N γω 1 d c/n N = Γ Nb a 0 (N). Then, if f S k (Γ 0 (N)), for all γ Γ 0 (N), f ωn γ = f γ ω N = f ωn = f ωn S k (Γ 0 (N)). Now, the operator ω N is an involution (f ω 2 = f) and an endomorphism N of the space of cusp forms, so it only has eigenvalues ±1, hence S k (Γ 0 (N)) = S + k (Γ 0(N)) S k (Γ 0(N)), where S ± k (Γ 0(N)) are the eigenspaces corresponding to ±1 respectively. Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

10 The functional equation Then, if f S k (Γ 0 (N)), we have f = ±f ωn f(z) = ±N k/2 z k f This modular relation implies the functional equation where, as usual, Λ(s, f) = ±i k Λ(k s, f), Λ(s, f) = ( ) N s Γ(s)L(s, f) 2π and L(s, f) is the Dirichlet series associated to f. ( 1 ). Nz Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

11 Hecke s converse theorem Let now N = 1, 2, 3, 4. Consider the Dirichlet series L(s) = a n n s, convergent in a right half plane, and assume the functional equation Λ(s) = ±i k Λ(k s), where Λ(s) = Then, f S k (Γ 0 (N)), where f(z) = a n e(nz). ( ) N s Γ(s)L(s). 2π Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

12 Hecke s converse theorem The result is due to the fact that, for these values of the level N, Γ 0 (N) = T, W N, with W N = ω N T 1 ω N. In fact, by Mellin inversion formula, the functional equation gives ( ) i f(iy) = ±i k N k/2 y k f f = ±f Ny ωn. Then, Fourier expansion = f T = f. f ωn = ±f = f WN = f. Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

13 What about N 5? the number of generators of Γ 0 (N) grows. the space of L-function satisfying the functional equation has infinite dimension. the space of modular/cusp forms for Γ 0 (N) is always finite dimensional. the single functional equation is not enough to conclude. we need additional conditions on the Dirichlet series. Two main approaches to the matter: the twist-theory: assume that proper twists by Dirichlet characters of the original L-function satisfy a functional equation. (Weil) Conrey-Farmer s converse theorem: besides the usual functional equation assume the existence of a proper Euler product. Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

14 Modular forms and Dirichlet characters Definition A Dirichlet character modulo N 1 is a group homomorphism ψ : Z C periodic of period N, such that ψ(a) 0 iff (N, a) = 1. So, let ψ be a Dirichlet character modulo N, M k (Γ 0 (N), ψ) is the space of the modular forms f M k (Γ 1 (N)) such that ( ) a b for all γ = Γ c d 0 (N) f γ = ψ(d)f. The following decompositions hold M k (Γ 1 (N)) = S k (Γ 1 (N)) = ψ (mod N) ψ (mod N) M k (Γ 0 (N), ψ) S k (Γ 0 (N), ψ) Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

15 The functional equation Let f S k (Γ 0 (N), ψ) and let g = f ωn. Then g γ = f ωn γ = f γ ω N = ψ(a)f ωn = ψ(d)g = g S k (Γ 0 (N), ψ). So, f, g are related by equation f(i/y) = N k 2 i k g(iy/n), which, with the usual analytic technique, leads to the functional equation Λ(s, f) = i k Λ(k s, g), where Λ(s, f) = Λ(s, g) = ( ) N s Γ(s)L(s, f) L(s, f) = 2π ( ) N s Γ(s)L(s, g) L(s, g) = 2π a n e(nz) b n e(nz). Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

16 The twist theory Let N, D be coprime integers, ψ a character modulo N, χ a primitive character modulo D and f S k (Γ 0 (N), ψ). f(z) = a n e(nz) f χ (z) = a n χ(n)e(nz). It can be proved that f χ S k (Γ 0 (ND 2 ), ψχ 2 ) and if g = f ωn Λ 1 (s, χ) = i k ω(χ)λ 2 (k s, χ), where ω(χ) = ψ(d)χ(n)τ 2 χd 1, τ χ is the Gauss sum, L 1 (s, χ) = a n χ(n)n s L 2 (s, χ) = b n χ(n)n s ( ) D N s ( ) D N s Λ 1 (s, χ) = Γ(s)L 1 (s, χ) Λ 2 (s, χ) = Γ(s)L 2 (s, χ) 2π 2π and (b n ) is the sequence of Fourier coefficients of g at. Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

17 Weil s converse theorem Assumptions: (a n ), (b n ), n 1 with polynomial growth. S finite set of primes, including those dividing N. For D = 1 or D / S, for any primitive character χ (mod D), Λ 1 (s, χ), Λ 2 (s, χ) are EBV and satisfy Λ 1 (s, χ) = i k ω(χ)λ 2 (k s, χ). Then Moreover, f(z) = a n e(nz) S k (Γ 0 (N), ψ) g(z) = b n e(nz) S k (Γ 0 (N), ψ). g = f ωn. Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

18 Weil s converse theorem: sketch of the proof By Mellin inversion formula, using the functional equation for twists, f χ = ω(χ)g χ 0 1/(D2 N) 1 0 so f χ 0 1 D 2 N 0 = ω(χ)g χ. In particular, choosing D = 1, we get g = f ωn. For D, s / S distinct odd primes g D r Nm s = ψ(s)g where Ds rnm = 1. Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

19 Weil s converse theorem: sketch of the proof We want to prove that for a, b, c, d Z such that ad bnc = 1 By Dirichlet s theorem g a Nc b d = ψ(d)g. Theorem Given m, n coprime integers, the arithmetic progression contains infinitely many primes. m, m + n, m + 2n,..., m + kn,... (a, Nc) = (d, Nc) = 1, so there exist u, v such that D = a unc s = d vnc and D, s / S are distinct primes, since S is a finite set. Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

20 Weil s converse theorem: sketch of the proof Taking r = av + uvnc + b ud, we have ( ) ( ) ( ) ( ) a b 1 u D r 1 v = Nc d 0 1 Nc s 0 1 hence g a Nc b d = g 1 u 0 1 D Nc r s 1 v 0 1 = g D Nc r s. Then we conclude g a Nc b d = ψ(s)g = ψ(d)g. Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

21 Conrey-Farmer s converse theorem L(s) = a n n s and f(z) = a n e(nz) L satisfies the functional equation ( ) N s Λ(s) = Γ(s)L(s) = ±i k Λ(k s) 2π L has an Euler product of the form L(s) = p L p (s) L p (s) = (1 a p p s + p k 1 2s ) 1 if p N L p (s) = (1 p k/2 1 s ) 1 if p N L p (s) = 1 if p 2 N Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

22 Conrey-Farmer s converse theorem Then, for 5 N 17 and N = 23, we have f S k (Γ 0 (N)). Hecke and Atkin-Lehner operators for p prime T p = ( ) p T p, U p : S k (Γ 0 (N)) S k (Γ 0 (N)) p 1 + a=0 ( ) 1 a 0 p the Euler product is equivalent to and U p = a=0 ( ) p a 0 p T p f = a p f, U p f = f p 0 0 1, U p f = 0 respectively for p N, p N and p 2 N. Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

23 Conrey-Farmer s converse theorem For any value of the level N f T = f (by the Fourier expansion) f ωn = ±f (by the functional equation) = f WN = f. Let N = 5, 7, 9. In these cases Γ 0 (N) = W N, T, M 2 where M 2 = ( ) 2 1. N (N + 1)/2 2 N = T2 f = a 2 f. combining with f T = f and f ωn = ±f, we get f M2 = f. then for all γ Γ0 (N), f γ = f. for N = 6, 8, 10, 12, 16 we again only use the local factor at p = 2, distinguishing 2 N and 4 N. Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

24 Conrey-Farmer s converse theorem let N = 11, 14, 15, 17, 23. In this cases, the key point is Lemma Let γ SL 2 (Z) satisfy f (1 γ)ε = f (1 γ), where ε SL 2 (R) is an elliptic matrix of infinite order. Then, f γ = f. We recall that a matrix ε SL 2 (R) is elliptic iff tr(ε) < 2. Equivalently, its eigenvalues have absolute value 1, but they are not roots of unity. for N = 13, the converse theorem has been proved in a more recent paper (2006), assuming the usual functional equation and the Euler product for the local factors at the primes p = 2, 3. Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25

25 Thank you for your attention! Giamila Zaghloul (DIMA unige) Converse theorems 10 novembre / 25