Open Access

Half-integral weight p-adic coupling of weakly holomorphic and holomorphic modular forms

Research in Number Theory20151:26

https://doi.org/10.1007/s40993-015-0027-1

Received: 17 June 2015

Accepted: 5 October 2015

Published: 21 December 2015

Abstract

In this paper, we consider p-adic limits of \(\beta ^{-n}g|U_{p^{2}}^{n}\) for half-integral weight weakly holomorphic Hecke eigenforms g with eigenvalue λ p =β+β under \(T\phantom {\dot {i}\!}_{p^{2}}\) and prove that these equal classical Hecke eigenforms of the same weight. This result parallels the integral weight case, but requires a much more careful investigation due to a more complicated structure of half-integral weight weakly holomorphic Hecke eigenforms.

Keywords

Weakly holomorphic modular forms p-adic congruences Half-integral weight Hecke eigenforms Shimura lift Shintani lift