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arXiv · 2307.07438

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

Scott Ahlgren·Nickolas Andersen·Robert Dicks
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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 $\ell$.

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Scott Ahlgren, Nickolas Andersen, Robert Dicks. 2023-07-14. The Shimura lift and congruences for modular forms with the eta multiplier. https://arxiv.org/abs/2307.07438

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