The Shimura lift and congruences for modular forms with the eta multiplier
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917654928818176 |
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| author | Ahlgren, Scott Andersen, Nickolas Dicks, Robert |
| author_facet | Ahlgren, Scott Andersen, Nickolas Dicks, Robert |
| contents | 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 $λ+1/2$ and level $N$ has weight $2λ$ 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 $\ell$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07438 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Shimura lift and congruences for modular forms with the eta multiplier Ahlgren, Scott Andersen, Nickolas Dicks, Robert Number Theory 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 $λ+1/2$ and level $N$ has weight $2λ$ 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 $\ell$. |
| title | The Shimura lift and congruences for modular forms with the eta multiplier |
| topic | Number Theory |
| url | https://arxiv.org/abs/2307.07438 |