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On mod p modular forms of weight one

CNTA/ACTN VII, Montreal, May 25 Mai, 2002

Bas Edixhoven Université de Rennes 1

Universiteit Leiden (from september 1)

http://www.maths.univ-rennes1.fr
edix@univ-rennes1.fr

Aim of the talk:

to convince you that mod p modular forms (and especially weight one forms modulo 2) deserve to be studied, and that they are not scary. to convince you that mod p modular forms (and especially weight one forms modulo

2) deserve to be studied, and that they are not scary.

Importance: representations G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) to convince you that mod p modular forms (and especially weight one forms modulo

2) deserve to be studied, and that they are not scary.

Importance: representations G_{ℚ} → GL_{2}(\overline{𝔽}_{p})

Why not scary: one can compute with them quite easily, without doing algebraic geometry but just group cohomology (even for weight one).

Serre’s conjecture (1987):

Let p be prime, ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) continuous, irreducible and odd.

Then ρ comes from a mod p modular form of type (N(ρ), k(ρ), ε(ρ)).

Let p be prime, ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) continuous, irreducible and odd.

Then ρ comes from a mod p modular form of type (N(ρ), k(ρ), ε(ρ)).

N(ρ) (the level) is the Artin conductor away from p.

k(ρ) (the weight) is given by a complicated recipe in terms of ρ|I_{p}.

ε(ρ): (ℤ/N(ρ)ℤ)^{∗} → \overline{𝔽}_{p}^{∗} given by:

det ∘ρ = ε(ρ)χ^{k(ρ)−1} .

N(ρ) and k(ρ) are meant to be minimal.

What are mod p modular forms? For N ≥ 5, p not dividing N, and k integer one defines, à la Katz: S(N, k)_{𝔽_{p}} = H^{0}(X_{1}(N)_{𝔽_{p}}, ω^{⊗k}(−cusps )).

What are mod p modular forms?

For N ≥ 5, p not dividing N, and k integer one defines, à la Katz:

S(N, k)_{𝔽_{p}} = H^{0}(X_{1}(N)_{𝔽_{p}}, ω^{⊗k}(−cusps )).

S(N, k)_{𝔽_{p}} is a finite dimensional 𝔽_{p}-vector space, equipped with Hecke operators T_{n} (n ≥ 1) and diamond operators ⟨a⟩ (a in (ℤ/Nℤ)^{∗}).

q-expansions: S(N, k)_{𝔽_{p}} ↪ 𝔽_{p}[[q]].

What are mod p modular forms?

For N ≥ 5, p not dividing N, and k integer one defines, à la Katz:

S(N, k)_{𝔽_{p}} = H^{0}(X_{1}(N)_{𝔽_{p}}, ω^{⊗k}(−cusps )).

S(N, k)_{𝔽_{p}} is a finite dimensional 𝔽_{p}-vector space, equipped with Hecke operators T_{n} (n ≥ 1) and diamond operators ⟨a⟩ (a in (ℤ/Nℤ)^{∗}).

q-expansions: S(N, k)_{𝔽_{p}} ↪ 𝔽_{p}[[q]].

To f = ∑ a_{n}(f)q^{n} in S(N, k)_{\overline{𝔽}_{p}} a normalized eigenform one can associate a semisimple continuous

ρ_{f}: G_{ℚ} → GL_{2}(\overline{𝔽}_{p})

(Deligne).

Naive mod p modular forms. One may reduce characteristic zero modular forms mod p:

S(N, k)_{ℤ[1/N]} → S(N, k)_{𝔽_{p}}.

Naive mod p modular forms. One may reduce characteristic zero modular forms mod p:

S(N, k)_{ℤ[1/N]} → S(N, k)_{𝔽_{p}}. One has: S(N, k)_{ℤ[1/N]} =

{f ∈ S(N, k)_{ℂ} | a_{n}(f) ∈ ℤ[1/N]}

Naive mod p modular forms.

One may reduce characteristic zero modular forms mod p:

S(N, k)_{ℤ[1/N]} → S(N, k)_{𝔽_{p}}.

One has: S(N, k)_{ℤ[1/N]} =

{f ∈ S(N, k)_{ℂ} | a_{n}(f) ∈ ℤ[1/N]}

Unfortunately: S(N, k)_{ℤ[1/N]} → S(N, k)_{𝔽_{p}} is not surjective for k = 1.

Naive mod p modular forms.

One may reduce characteristic zero modular forms mod p:

S(N, k)_{ℤ[1/N]} → S(N, k)_{𝔽_{p}}.

One has: S(N, k)_{ℤ[1/N]} =

{f ∈ S(N, k)_{ℂ} | a_{n}(f) ∈ ℤ[1/N]}

Unfortunately: S(N, k)_{ℤ[1/N]} → S(N, k)_{𝔽_{p}} is not surjective for k = 1.

Even worse: S(N, k, ε̃)_{\overline{ℤ}_{p}} → S(N, k, ε)_{\overline{𝔽}_{p}} is not always surjective even if k > 1. (Carayol’s lemma.)

Conclusion:

Serre’s conjecture comes out much more natural if one considers mod p modular forms à la Katz.

For example: k(ρ) = 1 if and only if ρ is unramified at p.

Let p be prime, ρ: G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) continuous, irreducible and odd.

Then ρ comes from a mod p modular form of type (N(ρ), k(ρ), ε(ρ)).

What is known, and what is not.

Theorem (SMRGCECVD...) If ρ is modular, then Serre’s conjecture is true for ρ, except possibly if p = 2, ρ is unramified at 2 and ρ(Frob _{2}) has a double eigenvalue.

Theorem (SMRGCECVD...) If ρ is modular, then Serre’s conjecture is true for ρ, except possibly if p = 2, ρ is unramified at 2 and ρ(Frob _{2}) has a double eigenvalue.

In the last case, one only does not know if ρ comes from a form of weight one (it does come from weight two).

Conversely, it is not known if ρ_{f} is unramified at two for f mod 2 of weight one.

Theorem (SMRGCECVD...) If ρ is modular, then Serre’s conjecture is true for ρ, except possibly if p = 2, ρ is unramified at 2 and ρ(Frob _{2}) has a double eigenvalue.

In the last case, one only does not know if ρ comes from a form of weight one (it does come from weight two).

Conversely, it is not known if ρ_{f} is unramified at two for f mod 2 of weight one.

Is ρ modular? Ask Ramakrishna and Taylor. (A p-adic lift of ρ is potentially modular.)

Why one cannot always lift in weight one.

On the modular forms side:

S(N, 1)_{𝔽_{p}} = H^{0}(X_{1}(N)_{𝔽_{p}}, ω(−cusps )).

The degree deg (ω(−cusps )) is about half the genus of X_{1}(N).

Why one cannot always lift in weight one.

On the modular forms side:

S(N, 1)_{𝔽_{p}} = H^{0}(X_{1}(N)_{𝔽_{p}}, ω(−cusps )).

The degree deg (ω(−cusps )) is about half the genus of X_{1}(N).

On the Galois side: if f̃ lifts f, then have

ρf̃: G_{ℚ} → GL_{2}(ℂ)

lifting ρ_{f}.

But then the image of ρf̃ in PGL_{2}(ℂ) is very small.

From weight one to weight p. Hasse:

S(N, 1)_{𝔽_{p}} → S(N, p)_{𝔽_{p}}, f ↦ Af. On q-expansions:

∑ a_{n}q^{n} ↦ ∑ a_{n}q^{n}.

From weight one to weight p. Hasse:

S(N, 1)_{𝔽_{p}} → S(N, p)_{𝔽_{p}}, f ↦ Af. On q-expansions:

∑ a_{n}q^{n} ↦ ∑ a_{n}q^{n}.

Frobenius:

S(N, 1)_{𝔽_{p}} → S(N, p)_{𝔽_{p}}, f ↦ f^{p}. On q-expansions:

∑ a_{n}q^{n} ↦ ∑ a_{n}q^{pn}.

Each leads to an algorithm.

The Hasse algorithm. Suppose N ≥ 5 is odd. Let 𝕋 be the Hecke algebra of S(2N, 2)_{ℂ}. Put:

M := (𝕋^{∨}/𝕋^{∨} ∩ w_{2}𝕋^{∨})_{2}. Then 4M = 0, and:

S(N, 1)_{𝔽_{2}} = M/M[2], as Hecke modules (except for T_{2}).

The Hasse algorithm. Suppose N ≥ 5 is odd. Let 𝕋 be the Hecke algebra of S(2N, 2)_{ℂ}. Put:

M := (𝕋^{∨}/𝕋^{∨} ∩ w_{2}𝕋^{∨})_{2}. Then 4M = 0, and:

S(N, 1)_{𝔽_{2}} = M/M[2], as Hecke modules (except for T_{2}).

Principle of proof: geometry over ℤ_{2}. Motivation from a question by Agashe and Stein (comparison of two lattices).

The Hasse algorithm.

Suppose N ≥ 5 is odd. Let 𝕋 be the Hecke algebra of S(2N, 2)_{ℂ}. Put:

M := (𝕋^{∨}/𝕋^{∨} ∩ w_{2}𝕋^{∨})_{2}.

Then 4M = 0, and:

S(N, 1)_{𝔽_{2}} = M/M[2],

as Hecke modules (except for T_{2}).

Principle of proof: geometry over ℤ_{2}.

Motivation from a question by Agashe and Stein (comparison of two lattices).

Note: 𝕋 can be computed using modular symbols.

The Frobenius algorithm. Let N ≥ 5 and p a prime not dividing N. Let 𝕋 be the Hecke algebra of S(N, p)_{ℂ}. Put B := \frac{p+2}{24} N^{2} ∏_{l|N}(1 − l^{−2}). Then:

S(N, 1)_{𝔽_{p}} =

{f: 𝕋 → 𝔽_{p} | f(T_{n}) = 0, for p |̸n ≤ B }, as Hecke modules.

The Frobenius algorithm. Let N ≥ 5 and p a prime not dividing N. Let 𝕋 be the Hecke algebra of S(N, p)_{ℂ}. Put B := \frac{p+2}{24} N^{2} ∏_{l|N}(1 − l^{−2}). Then:

S(N, 1)_{𝔽_{p}} =

{f: 𝕋 → 𝔽_{p} | f(T_{n}) = 0, for p |̸n ≤ B }, as Hecke modules.

Principle of proof:

qd/dq: S(N, p)_{𝔽_{p}} → S(N, p + 2)_{𝔽_{p}}, and a result of Jacob Sturm (for B).

Modular symbols modulo p. In the Frobenius algorithm, one only needs to compute 𝔽_{p} ⊗ 𝕋.

Modular symbols modulo p.

In the Frobenius algorithm, one only needs to compute 𝔽_{p} ⊗ 𝕋.

Theorem H^{1}(X_{1}(N)(ℂ), 𝔽_{2}) is a faithful 𝔽_{2} ⊗ 𝕋-module.

The proof uses group schemes over 𝔽_{2}, Dieudonné modules, and de Rham cohomology.

Modular symbols modulo p.

In the Frobenius algorithm, one only needs to compute 𝔽_{p} ⊗ 𝕋.

Theorem H^{1}(X_{1}(N)(ℂ), 𝔽_{2}) is a faithful 𝔽_{2} ⊗ 𝕋-module.

The proof uses group schemes over 𝔽_{2}, Dieudonné modules, and de Rham cohomology.

Theorem For 2 ≤ k < p,

H_{par}^{1}(Γ_{1}(N), Sym ^{k−2}(𝔽_{p}^{2}))

is a faithful 𝔽_{p} ⊗ 𝕋(N, k)-module.

The proof uses Fontaine-Laffaille theory. One should be able to treat the important case k = p.

A result on eigenspaces. Let N ≥ 1, p prime not dividing Nx. Let V be a common non-zero eigenspace in S(N, p)_{\overline{𝔽}_{p}} for all T_{l}, l ≠ p prime, with system of eigenvalues a.

A result on eigenspaces. Let N ≥ 1, p prime not dividing Nx. Let V be a common non-zero eigenspace in S(N, p)_{\overline{𝔽}_{p}} for all T_{l}, l ≠ p prime, with system of eigenvalues a. Then V is of dimension at most two. If dim (V) = 1, then a does not occur in S(N, 1)_{\overline{𝔽}_{p}}. If dim (V) = 2, then the a-eigenspace V_{1} in S(N, 1)_{\overline{𝔽}_{p}} is of dimension one, with T_{p} eigenvalue the trace of T_{p} on V.

Computations.

There are published methods to compute S(N, 1)_{ℂ}.

Kiming, On the experimental verification of the Artin conjecture for 2-dimensional odd Galois representations over ℚ. In Lecture Notes in Mathematics 1585 (1994).

As far as I know, there are no published tables of mod p modular forms of weigt 1. But there is a letter from Mestre to Serre (october 1987).