## Abstract

Let F be a newform of weight 2k on Γ (N) with an odd integer N and a positive integer k, and ℓ be a prime larger than or equal to 5 with (ℓ, N) = 1. For each fundamental discriminant D, let χ_{D} be a quadratic character associated with quadratic field Q(D). Assume that for each D, the ℓ-adic valuation of the algebraic part of L(F⊗ χ_{D}, k) is non-negative. Let Wℓ+ (resp. Wℓ-) be the set of positive (resp. negative) fundamental discriminants D with (D, N) = 1 such that the ℓ-adic valuation of the algebraic part of L(F⊗ χ_{D}, k) is zero. We prove that for each sign ϵ, if Wℓϵ is a non-empty finite set, then Wℓϵ⊂{1,(-1)ℓ-12ℓ}.By this result, we prove that if ϵ is the sign of (- 1) ^{k}, then k≥ℓ-1ork=ℓ-12.These are applied to obtain a lower bound for #{D∈Wℓϵ:|D|≤X} and the indivisibility of the order of the Shafarevich–Tate group of an elliptic curve over Q. To prove these results, first we refine Waldspurger’s formula on the Shimura correspondence for general odd levels N. Next we study mod ℓ modular forms of half-integral weight with few non-vanishing coefficients. To do this, we use the filtration of mod ℓ modular forms and mod ℓ Galois representations.

Original language | English |
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Article number | 65 |

Journal | Research in Mathematical Sciences |

Volume | 9 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2022 Dec |

### Bibliographical note

Funding Information:The authors appreciate Ken Ono for his kind and helpful comments. The authors also appreciate a referee for careful reading and useful comments. These comments improved the previous version of this paper. The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1A2C1007517). The second author was supported by a KIAS Individual Grant (MG086301) at Korea Institute for Advanced Study.

Publisher Copyright:

© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

## Keywords

- Central values of modular L-functions
- Galois representations
- Mod ℓ
- Shimura correspondence

## ASJC Scopus subject areas

- Theoretical Computer Science
- Mathematics (miscellaneous)
- Computational Mathematics
- Applied Mathematics