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

Scott Ahlgren, Nickolas Andersen, Robert Dicks

Research output: Contribution to journalArticlepeer-review


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 ℓ which give rise to quadratic congruences modulo arbitrary powers of ℓ.

Original languageEnglish (US)
Article number109655
JournalAdvances in Mathematics
StatePublished - Jun 2024


  • Congruences for modular forms
  • Eta multiplier
  • Shimura lift

ASJC Scopus subject areas

  • General Mathematics


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