[Paper Review] Hasse invariant and group cohomology

Bas Edixhoven, Chandrashekhar Khare|ArXiv.org|Oct 25, 2002

Algebraic Geometry and Number Theory9 references10 citations

TL;DR

This paper establishes a group cohomology framework to emulate the role of the Hasse invariant in producing congruences between modular forms of weight 2 and weight $p+1$ modulo $p$, using degeneracy maps and Hecke eigenvalue lifting. The key contribution is a cohomological mechanism that resolves a puzzle in the Eichler-Shimura isomorphism by showing that weight 2 Hecke eigenforms lift to higher-weight forms via symmetric powers, extending to imaginary quadratic and totally real fields where geometric Hasse invariants are unavailable.

ABSTRACT

Let p be a prime number. The Hasse invariant is a modular form modulo p that is often used to produce congruences between modular forms of different weights. We show how to produce such congruences between forms of weights 2 and p+1, in terms of group cohomology. We also show how our method works in the contexts of quadratic imaginary fields (where there is no Hasse invariant available) and Hilbert modular forms over totally real fields of even degree.

Motivation & Objective

• To resolve a cohomological puzzle: why Hecke eigenvalues in $H^1(\Gamma_1(N), \mathbb{F}_p)$ also arise from $H^1(\Gamma_1(N), \mathrm{Sym}^{p-1}(\mathbb{F}_p^2))$ despite the irreducibility of the symmetric power representation.
• To develop a group cohomology substitute for the Hasse invariant in contexts where the geometric Hasse invariant does not exist, such as over imaginary quadratic fields and Hilbert modular forms over totally real fields.
• To extend the method of weight-raising congruences from classical modular forms to higher-rank settings, including non-parallel weight Hilbert modular forms.
• To provide a cohomological interpretation of level-raising criteria for $p$-newforms in various weights, particularly for $2 < k \leq p+1$ and $k > p+1$.

Proposed method

• Use the degeneracy map $\alpha: H^1(\Gamma_1(N), \mathbb{F}_p)^2 \to H^1(\Gamma_1(N) \cap \Gamma_0(p), \mathbb{F}_p)$, defined as the sum of restriction and twisted restriction via conjugation by $\begin{pmatrix} p & 0 \\ 0 & 1 \end{pmatrix}$, to relate cohomology groups of different levels.
• Apply the Eichler-Shimura isomorphism to identify modular forms with cohomology classes in $H^1(\Gamma_1(N), \mathbb{F}_p)$ and $H^1(\Gamma_1(N), \mathrm{Sym}^{p-1}(\mathbb{F}_p^2))$, enabling comparison of Hecke eigenvalues.
• Leverage the injectivity of the degeneracy map (Lemma 1) and strong approximation in the totally real case to control the kernel and ensure eigenvalue lifting.
• Use the Jacquet-Langlands correspondence in the Hilbert modular form setting to transfer cohomological data between automorphic representations and forms on quaternion algebras.
• Construct a representation $\mathrm{Sym}^{p-1}(\mathbb{F}_\wp^2) \otimes \mathrm{Sym}^{p-1}(\mathbb{F}_\wp^2)^\sigma \otimes \cdots$ that appears as a direct summand in the induced representation from the Borel subgroup, enabling eigenvalue transfer.
• Apply results from [K] to derive level-raising criteria for $p$-newforms, distinguishing cases based on weight $k$ relative to $p+1$.

Experimental results

Research questions

• RQ1 Why do Hecke eigenvalues in $H^1(\Gamma_1(N), \mathbb{F}_p)$ also arise from $H^1(\Gamma_1(N), \mathrm{Sym}^{p-1}(\mathbb{F}_p^2))$ despite the irreducibility of the symmetric power representation?
• RQ2 Can a cohomological substitute for the Hasse invariant be constructed in settings where the geometric Hasse invariant does not exist, such as over imaginary quadratic fields?
• RQ3 How can weight-raising congruences between modular forms of weights 2 and $p+1$ be systematically described using group cohomology?
• RQ4 What is the cohomological mechanism underlying the lifting of $p$-newforms from level $N$ to level $Np$ in different weight ranges?
• RQ5 How does the degeneracy map interact with Hecke operators to preserve eigenvalue systems across different cohomology groups?

Key findings

• The degeneracy map $\alpha: H^1(\Gamma_1(N), \mathbb{F}_p)^2 \to H^1(\Gamma_1(N) \cap \Gamma_0(p), \mathbb{F}_p)$ is injective, providing a key technical tool for lifting cohomology classes and eigenvalues.
• A weight 2 Hecke eigenform modulo $p$ with eigenvalues in $\overline{\mathbb{F}_p}$ also arises from a form in $H^1(\Gamma_1(N), \mathrm{Sym}^{p-1}(\mathbb{F}_p^2))$, resolving the cohomological puzzle.
• For $k=2$, a newform $f$ is congruent to a $p$-newform modulo $\wp$ if and only if $a_p(f)^2 \equiv \varepsilon_f(p) \pmod{\wp}$, where $\varepsilon_f$ is the nebentypus character.
• For $2 < k \leq p+1$, $f$ is congruent to a $p$-newform modulo $\wp$ if and only if $a_p(f) \equiv 0 \pmod{\wp}$.
• For $k > p+1$, every newform $f$ in $S_k(\Gamma_1(N))$ is congruent to a $p$-newform in $S_k(\Gamma_1(N) \cap \Gamma_0(p))$, indicating that level-raising is always possible in higher weights.
• In the setting of Hilbert modular forms over a totally real field $F$ of even degree, an irreducible Galois representation $\rho: G_F \to \mathrm{GL}_2(\overline{\mathbb{F}_p})$ arising from weight $(2,\dots,2)$ also arises from weight $(p+1,\dots,p+1)$, via the Jacquet-Langlands correspondence and cohomological lifting.