Quadratic Congruences for half-integral weight cusp forms with the eta multiplier

TL;DR

This paper proves new quadratic congruences for half-integral weight cusp forms with eta multiplier, extending previous results to arbitrary characters using modular Galois representations and their large image properties.

Contribution

It generalizes quadratic congruences for half-integral weight cusp forms to all characters, employing advanced Galois representation techniques.

Findings

Quadratic congruences hold for arbitrary characters.

Existence of Galois elements with prescribed conjugacy classes.

Large image properties of modular Galois representations.

Abstract

Let $\ell \geq 5$ be a prime, and let $\nu_\eta$ denote the Dedekind eta multiplier. For an odd integer $r$, and a real Dirichlet character $\psi$, recent work of Ahlgren, Andersen, and the author showed that quadratic congruences modulo $\ell$ hold for a wide range of half-integral weight cusp forms with multiplier $\psi\nu_\eta^r$, vastly generalizing certain congruences discovered by Atkin for the partition function. In this paper, we show that such congruences hold when $\psi$ is an arbitrary character. Our methods rely on the theory of modular Galois representations. For primes $\ell \geq 5$, the core of our work is the study of modular Galois representations modulo $\ell$ attached to integer-weight eigenforms with arbitrary Nebentypus whose images are large in a precise sense. Our key new result is that, given a finite set of such representations and $\gamma \in \SL_2(\F_\ell)$, there exists $\sigma \in \Gal(\bar{\Q}/\Q(\zeta_\ell))$ whose images under the representations are in the conjugacy class of $\gamma^2$.