The Shimura lift and congruences for modular forms with the eta multiplier

TL;DR

This paper extends the Shimura correspondence to certain half-integral weight modular forms with eta multiplier, revealing new arithmetic properties and congruences related to primes and Atkin-Lehner eigenvalues.

Contribution

It establishes a Shimura-type lift for eta-multiplier twisted forms, detailing their level, weight, and eigenvalues, and demonstrates the existence of infinitely many primes producing quadratic congruences.

Findings

The lift increases weight from λ+1/2 to 2λ and level from N to 6N.

The lift is new at primes 2 and 3 with specific eigenvalues.

Infinitely many primes induce quadratic congruences modulo powers of these primes.

Abstract

The Shimura correspondence is a fundamental tool in the study of half-integral weight modular forms. In this paper, we prove a Shimura-type correspondence for spaces of half-integral weight cusp forms which transform with a power of the Dedekind eta multiplier twisted by a Dirichlet character. We prove that the lift of a cusp form of weight $\lambda+1/2$ and level $N$ has weight $2\lambda$ and level $6N$, and is new at the primes $2$ and $3$ with specified Atkin-Lehner eigenvalues. This precise information leads to arithmetic applications. For a wide family of spaces of half-integral weight modular forms we prove the existence of infinitely many primes $\ell$ which give rise to quadratic congruences modulo arbitrary powers of $\ell$.