Weight reduction for cohomological mod $p$ modular forms over imaginary quadratic fields

TL;DR

This paper proves a weight reduction result for cohomological mod p modular forms over imaginary quadratic fields, showing that eigenvalues can be realized in simpler coefficient systems under certain conditions.

Contribution

It establishes a new weight reduction theorem for cohomological mod p modular forms over imaginary quadratic fields, extending previous results to this setting.

Findings

Eigenvalues in cohomology with coefficients in V also appear in simpler coefficient systems.

The weight reduction holds for all but a few exceptional cases.

The result applies to inert primes greater than 5 coprime with the level ideal.

Abstract

Let $ F$ be an imaginary quadratic field and $\mathcal{O}$ its ring of integers. Let $ \mathfrak{n} \subset \mathcal{O} $ be a non-zero ideal and let $ p> 5$ be a rational inert prime in $F$ and coprime with $\mathfrak{n}$. Let $ V$ be an irreducible finite dimensional representation of $ \bar{\mathbb{F}}_{p}[{\rm GL}_2(\mathbb{F}_{p^2})].$ We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in $ V$ already lives in the cohomology with coefficients in $ \bar{\mathbb{F}}_{p}\otimes det^e$ for some $ e \geq 0$; except possibly in some few cases.