Download Number of non-zero coefficients of modular forms

Number of non-zero coefficients of modular
forms modulo p
Analytic Aspects of Number Theory, Zürich, 2015
Joël Bellaı̈che
(first part is joint work with Kannan Soundararajan)
Brandeis University
May 18, 2015
Objectives
Let F be a finite field. For f =
P∞
n=0 an q
n
∈ F[[q]], define
ψ(f , x) = #{n ; 0 ≤ n ≤ x, an 6= 0}.
π(f , x) = #{` prime ; ` ≤ x, a` 6= 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 0 such that k 0 (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 6= 0 =⇒ gcd(n, Np) = 1.
Think of Fk (N, F) as the “new” part in Mk (N, F).
If f =
P
an q n ∈ Mk (N, F), then
X
f 0 :=
an q n ∈ Fk (N 2 , F).
(n,Np)=1
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 6= f ∈ Fk (N, F).
Serre (1976) : ψ(f , x) is lacunary.
x
(log x)α
Ahlgren (1999) : ψ(f , x) Chen (2012): ψ(f , x) for some α > 0. In particular, f
x
log 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 ` - Np, set
X
X
T` (
an q n ) =
(an` + `k−1 an/` )q n .
The T` ’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 = (λ` )`-Np , define
Fk (N, F)λ = {f ∈ Fk (N, F); ∀` ∃n (T` − λ` )n f = 0}.
M
Then Fk (N, F) =
Fk (N, F)λ (finite sum).
λ
Define the Hecke algebra Aλ as the closed subalgebra of
End(Fk (N, F)λ ) generated by the T` . It is local and complete.
Rough skecth of the proof :
the case of a generalized eigenform
Fix λ = (λ` ). Suppose f ∈ Fk (N, F)λ .
Two kinds of `:
(i) λ` = 0 or T` locally nilpotent on Fk (N, F)λ .
(ii) λ` 6= 0 or T` invertible or T` ∈ A∗λ (density α(f )).
Define h(f ) (strict order of nilpotence) as the largest h for which
there exist `1 , . . . , `h of type (i) such that T`1 . . . T`h f 6= 0.
Counting n square-free such that an 6= 0. Write
n = `1 . . . `h `01 . . . `0s with `i of type (i), `0j of type (ii).
an (f )=a1 (T`0 ...T`0 T`1 ...T`h f )6=0⇐⇒h≤h(f ) and a1 (T`0 ...T`0 g )6=0,
1
s
1
s
where g = T`1 . . . T`h f (finitely many possible g ’s for a given f )
The T`0 ’s generate a subgroup H of A∗λ . T`01 . . . T`0s is a random
element of H (random walk). =⇒ count as if the `i were arbitrary
of type (ii) and multiply by the proportion of T ∈ H such that
a1 (Tg ) 6= 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
Pof weight −1/2 (say), e.g.
g (q) = η(q)−1 = q 1/24 p(n)q n modulo p. It is conjectured that
ψ(g , x) x. Very little is known about it: not known that
ψ(g , x) x 1/2+ for any > 0 .
k
Define fk as g 1−p . 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
for x Ak ,
log x
we could break the bound x 1/2 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
CountingPjust prime numbers ` such that a` 6= 0 is more promising.
For f =
an q n ∈ 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 6= 0, then 0 < δ(f ) < 1.
Corollary
P
P
Let f =
an q n , g = bn q n ∈ Fk (N, F). If a` = b` for a set of
density one of `, then f = g .
Counting non-zero coefficients at primes: uniformity?
Remember:
(2)
π(f , x) ∼ δ(f )
x
, with 0 < δ(f ) < 1.
log x
To get some uniformity result, like (?), from (2), we need
(a) control of δ(f ), namely that δ(f ) does not approach 0 (at
k
least for f = fk = g 1−p 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) =
L
λ Fk (N, F)λ
(finite sum).
Theorem (Deligne)
To λ is attached a unique semi-simple Galois representation
ρ̄ : GQ,Np → GL2 (F) such that tr ρ̄(Frob ` ) = λ` .
Let ad0 ρ̄ be the adjoint representation, of dimension 3.
Theorem
Assume that λ is such that ad0 ρ̄ is irreducible. Then there exist
c, c 0 such that 0 < c < c 0 < 1 and for all f ∈ F(N, F)λ ,
c < δ(f ) < c 0 .
Counting non-zero coefficients at primes: control of δ(f )
Theorem
Assume that λ is such that ad0 ρ̄ is irreducible. Then there exist
c, c 0 such that 0 < c < c 0 < 1 and for all f ∈ F(N, F)λ ,
c < δ(f ) < c 0 .
The projective image of ρ̄ 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 ρ̄ 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 ρ̄ is reducible? The theorem is false, but
P
Say f = an q n ∈ Fk (N, F)λ is abelian if a` depends only on `
(mod M) for some M, say f is dihedral if a` depends only on
Frob ` in a dihedral extension, special if it is linear combination of
abelian and dihedral forms.
Conjecture
Assume that λ is such that ad0 ρ̄ is reducible. (1) There exists
c, c 0 such that 0 < c < c 0 < 1 and for all non-special
f ∈ F(N, F)λ ,
c < δ(f ) < c 0 .
(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 (Bellaı̈che-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 6= f ∈ Fk (F), then 0 < δ(f ) < 1.
Theorem 2
Assume that λ is such that ad0 ρ̄ is irreducible. Then there exist
c, c 0 such that 0 < c < c 0 < 1 and for all f ∈ Fk (N, F)λ ,
c < δ(f ) < c 0 .
Remember the Hecke algebra Aλ acting on FkQ
(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 ` ) = T` and d = ωpk−1
Both Theorems 1 and 2 can be reduced to results about the image
of 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 ` ) = T` for every ` - Np.
As a topological algebra, Ak is generated by the T` ’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 6= 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
a` (f ) 6= 0 ⇐⇒ a1 (T` f ) 6= 0
(1)
⇐⇒ T` 6∈ Hf
(2)
⇐⇒ tr ρ(Frob ` ) 6∈ Hf
⇐⇒ Frob ` 6∈ (tr ρ)
−1
(3)
(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 ρ̄ is irreducible.
t
Consider tλ : GQ,Np → 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 ∩ SL12 (Aλ ).
Define following Pink Θ : M2 (Aλ ) → sl2 (Aλ ), x 7→ x − 12 tr (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.