Number of non-zero coefficients of modular forms modulo p Analytic - - PowerPoint PPT Presentation

Number of non-zero coefficients of modular forms modulo p

Analytic Aspects of Number Theory, Z¨ urich, 2015 Jo¨ el Bella¨ ıche

(first part is joint work with Kannan Soundararajan) Brandeis University

May 18, 2015

Objectives

Let F be a finite field. For f = ∞

n=0 anqn ∈ F[[q]], define

ψ(f , x) = #{n ; 0 ≤ n ≤ x, an = 0}. π(f , x) = #{ℓ prime ; ℓ ≤ x, aℓ = 0}. The objective is to give estimates of ψ(f , x), π(f , x) for x → ∞, when f is a modular form modulo p, (a) for f fixed; (b) uniformly in f (much harder).

Spaces of modular forms modulo p

Fix: a prime p an integer N ≥ 1 (the level), an element k ∈ Z/(p − 1)Z (the weight). Define Mk(N, Fp) ⊂ Fp[[q]] as the span of all reductions mod p of holomorphic modular forms with coefficients in Z, level Γ0(N), some weight k′ such that k′ (mod p − 1) = k. For F a finite extension of Fp, set Mk(N, F) = Mk(N, Fp) ⊗Fp F

Spaces of modular forms modulo p (continued)

Primes dividing Np, and their products, are annoying. Define Fk(N, F) as the subspace of Mk(N, F) of f such that an = 0 = ⇒ gcd(n, Np) = 1. Think of Fk(N, F) as the “new” part in Mk(N, F). If f = anqn ∈ Mk(N, F), then f ′ :=

• (n,Np)=1

anqn ∈ Fk(N2, F). It is no real restriction to consider only Fk(N, F).

Equivalent for ψ(f , x): previous results and statement

Assume p ≥ 3. Let 0 = f ∈ Fk(N, F). Serre (1976) : ψ(f , x) ≪

x (log x)α for some α > 0. In particular, f

is lacunary. Ahlgren (1999) : ψ(f , x) ≫

x log x .

Chen (2012): ψ(f , x) ≫ x(log log x)N

log x

for all N ≥ 1.

Theorem (B.-Soundararajan, 2014)

ψ(f , x) ∼ c(f )x(log log x)h(f ) (log x)α(f ) where c(f ) > 0, h(f ) ∈ N, α(f ) ∈ Q with 0 < α(f ) ≤ 3/4 are effectively computable constants.

Rough skecth of the proof : Hecke operators and Hecke algebras

Hecke operators: for ℓ ∤ Np, set Tℓ(

• anqn) =
• (anℓ + ℓk−1an/ℓ)qn.

The Tℓ’s stabilize Fk(N, F) and commute. But (Serre, Tate, Jochnowitz): only finitely many normalized eigenvectors in Fk(N, F) (though it is infinite-dimensional.) Use generalized eigenvectors instead. For λ = (λℓ)ℓ∤Np, define Fk(N, F)λ = {f ∈ Fk(N, F); ∀ℓ ∃n (Tℓ − λℓ)nf = 0}. Then Fk(N, F) =

• λ

Fk(N, F)λ (finite sum). Define the Hecke algebra Aλ as the closed subalgebra of End(Fk(N, F)λ) generated by the Tℓ. It is local and complete.

Rough skecth of the proof : the case of a generalized eigenform

Fix λ = (λℓ). Suppose f ∈ Fk(N, F)λ. Two kinds of ℓ: (i) λℓ = 0 or Tℓ locally nilpotent on Fk(N, F)λ. (ii) λℓ = 0 or Tℓ invertible or Tℓ ∈ A∗

λ (density α(f )).

Define h(f ) (strict order of nilpotence) as the largest h for which there exist ℓ1, . . . , ℓh of type (i) such that Tℓ1 . . . Tℓhf = 0. Counting n square-free such that an = 0. Write n = ℓ1 . . . ℓhℓ′

1 . . . ℓ′ s with ℓi of type (i), ℓ′ j of type (ii). an(f )=a1(Tℓ′

1...Tℓ′ s Tℓ1...Tℓhf )=0⇐

⇒h≤h(f ) and a1(Tℓ′

1...Tℓ′ s g)=0,

where g = Tℓ1 . . . Tℓhf (finitely many possible g’s for a given f ) The Tℓ′’s generate a subgroup H of A∗

λ. Tℓ′

1 . . . Tℓ′ s is a random

element of H (random walk). = ⇒ count as if the ℓi were arbitrary

• f type (ii) and multiply by the proportion of T ∈ H such that

a1(Tg) = 0. Then apply Landau’s method.

Uniformity for ψ(f , x)?

Why do we want uniformity? “To take the limit in f .” Let g be a modular form of weight −1/2 (say), e.g. g(q) = η(q)−1 = q1/24 p(n)qn modulo p. It is conjectured that ψ(g, x) ≍ x. Very little is known about it: not known that ψ(g, x) ≫ x1/2+ǫ for any ǫ > 0 . Define fk as g1−pk. The fk are true modular forms mod p. (1) ψ(fk, x) ∼ ck x(log log x)hk (log x)αk . If we could have (?) ψ(fk, x) ≫ x log x for x ≫ Ak, we could break the bound x1/2 for ψ(g, x). Unfortunately, there is no way to prove (?) with the method we proved (1) : ck goes to 0 very fast, error term is hopelessly large. To compute the equivalent for ψ(fk, x) we counted integers n with at least hk := h(fk) prime factors, so n ≥ 2hk and hk tends to grow exponentially in k, so 2hk grows double-exponentially.

Counting non-zero coefficients at primes: f fixed

Counting just prime numbers ℓ such that aℓ = 0 is more promising. For f = anqn ∈ Fk(N, F), that set is Frobenian (Serre 1976). So it has a density: π(f , x) ∼ δ(f ) x log x with 0 ≤ δ(f ) ≤ 1, δ(f ) ∈ Q.

Theorem

If f = 0, then 0 < δ(f ) < 1.

Corollary

Let f = anqn, g = bnqn ∈ Fk(N, F). If aℓ = bℓ for a set of density one of ℓ, then f = g.

Counting non-zero coefficients at primes: uniformity?

Remember: (2) π(f , x) ∼ δ(f ) x log x , with 0 < δ(f ) < 1. To get some uniformity result, like (?), from (2), we need (a) control of δ(f ), namely that δ(f ) does not approach 0 (at least for f = fk = g1−pk as above). (b) control of the error term (probably using GRH). Both are work in progress. We will discuss a result concerning δ(f ).

Counting non-zero coefficients at primes: control of δ(f )

Remember Fk(N, F) =

λ Fk(N, F)λ

(finite sum).

Theorem (Deligne)

To λ is attached a unique semi-simple Galois representation ¯ ρ : GQ,Np → GL2(F) such that tr ¯ ρ(Frob ℓ) = λℓ. Let ad0¯ ρ be the adjoint representation, of dimension 3.

Theorem

Assume that λ is such that ad0¯ ρ is irreducible. Then there exist c, c′ such that 0 < c < c′ < 1 and for all f ∈ F(N, F)λ, c < δ(f ) < c′.

Counting non-zero coefficients at primes: control of δ(f )

Theorem

Assume that λ is such that ad0¯ ρ is irreducible. Then there exist c, c′ such that 0 < c < c′ < 1 and for all f ∈ F(N, F)λ, c < δ(f ) < c′. The projective image of ¯ ρ can be abelian (in a torus), dihedral (in the normalizer of a torus), large (PSL2(Fq) or PGL2(Fq)) or exceptional (A4, S4 or A5). The condition ad0¯ ρ irreducible means that it is exceptional or large. The theorem should also be true for linear combinations of generalized eigenforms.

Counting non-zero coefficients at primes: reducible case

What happens when ad0¯ ρ is reducible? The theorem is false, but Say f = anqn ∈ Fk(N, F)λ is abelian if aℓ depends only on ℓ (mod M) for some M, say f is dihedral if aℓ depends only on Frob ℓ in a dihedral extension, special if it is linear combination of abelian and dihedral forms.

Conjecture

Assume that λ is such that ad0¯ ρ is reducible. (1) There exists c, c′ such that 0 < c < c′ < 1 and for all non-special f ∈ F(N, F)λ, c < δ(f ) < c′. (2) special forms are rare, i.e. killed by an ideal I in Aλ of codim ≥ 1. (2) is known for N = 1, p = 2 (Bella¨ ıche-Nicolas-Serre) and N = 1, p = 3 (Medvedowski). If f ∈ F(1, F2) is special, h = h(f ), then δ(f ) = 2−u(h)−v2(h)−1.

Interlude: pseudo-representations (of dim 2)

Let G be a group, A a ring. Representations ρ : G → GL2(A) don’t glue well. Replace then with their trace and determinant.

Definition (Chenevier)

A pseudo-rep. of dim 2 of G on A is a pair of maps t, d : G → A, s.t. t(1) = 2. d is a morphism from G to A∗ ∀g, h ∈ G, t(gh) = t(hg). ∀g, h ∈ G, t(gh) + d(h)t(gh−1) = t(g)t(h). If ρ is a representation, the pair t = tr ρ, d = det ρ is a pseudo-representation. When A is an algebraically closed field, the converse is true.

Theorems on δ(f ): what about proofs?

Theorem 1

If 0 = f ∈ Fk(F), then 0 < δ(f ) < 1.

Theorem 2

Assume that λ is such that ad0¯ ρ is irreducible. Then there exist c, c′ such that 0 < c < c′ < 1 and for all f ∈ Fk(N, F)λ, c < δ(f ) < c′. Remember the Hecke algebra Aλ acting on Fk(N, F)λ. Similarly, define Ak acting on Fk(N, F). One has Ak =

λ Aλ.

Proposition 1

There exists a pseudo-representation (t, d) of ρ : GQ,Np over Ak such that t(Frob ℓ) = Tℓ and d = ωk−1

p

Both Theorems 1 and 2 can be reduced to results about the image

• f t.

A result on t(GQ,Np)

Theorem 3

As a topological vector space, Ak is generated by t(GQ,Np). Indeed, t(GQ,Np) contains t(Frob ℓ) = Tℓ for every ℓ ∤ Np. As a topological algebra, Ak is generated by the Tℓ’s. But the closed subspace generated by t(GQ,Np) is an algebra, because t(g)t(h) = t(gh) + d(h)t(gh−1), and d(h) ∈ Fp; and 2 = t(1) ∈ t(GQ,Np). QED

Theorems about δ(f ): Theorem 3 implies Theorem 1

Let 0 = f ∈ Fk(F). The set of T ∈ Ak such that a1(Tf ) = 0 is a closed hyperplane of Ak. Let’s call it Hk. One has aℓ(f ) = 0 ⇐ ⇒ a1(Tℓf ) = 0 (1) ⇐ ⇒ Tℓ ∈ Hf (2) ⇐ ⇒ tr ρ(Frob ℓ) ∈ Hf (3) ⇐ ⇒ Frob ℓ ∈ (tr ρ)−1(Hf ) ⊂ GQ,Np. (4) Now, (tr ρ)−1(Hf ) is a closed subset of GQ,Np hence its complement is open and of positive Haar measure, and δ(f ) > 0 follows by Chebotarev.

Rough sketch of proof of Theorem 2

Let λ such that ad0¯ ρ is irreducible. Consider tλ : GQ,Np

t

→ Ak → Aλ and define dλ similarly. Then the pseudo-rep tλ, dλ comes from a representation ρ : GQ,Np → GL2(Aλ). Let G = ρ(GQ,Np) ⊂ GL2(Aλ). Let Γ = G ∩ SL1

2(Aλ).

Define following Pink Θ : M2(Aλ) → sl2(Aλ), x → x − 1

2tr (x)Id.

Define L(Γ) as the closed additive subgroup generated by Θ(Γ). Pink shows that L(Γ) is a Lie algebra over Zp.

Theorem 4

L(Γ) = sl2(P) for a closed pseudo-subring P of Aλ. Moreover FP = m, and Γ = Θ−1(sl2(P)). Theorem 4 reduces Theorem 2 to proving that the proportion of points of an affine space of dim d that lies in a suitable intersection of m quadrics is bounded away from 0 and below from 1, independently of d. The lower bound comes from a theorem of Warning, the upper from a computation involving Jacobi sums.