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

Abstract

In this paper, we consider p-adic limits of β −n g|U n p 2 for half-integral weight weakly holomorphic Hecke eigenforms g with eigenvalue λ p = β + β under T 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.

p-Adic properties of coefficients of certain half-integral weight modular forms

“…Our goal in this paper is to obtain similar p-adic statements to those in [6] for the parallel cases of Zagier's forms and Folsom-Ono's forms. Theorem 1.1.…”

“…Inspired by Zagier's work, Duke and Jenkins explore in [9] other half-integral weight weakly holomorphic modular forms in Kohnen's plus space of level 4. In [6], Bringmann, Guerzhoy, and Kane continue the study of these forms by investigating their p-adic properties, using a lifting procedure developed by Duke and Jenkins to link back to Zagier's original forms in order to give a p-adic relation between halfintegral weight weakly holomorphic modular forms and classical half-integral weight holomorphic modular forms. However, the approach used by Bringmann, Guerzhoy, and Kane only works for half-integral weight forms of weight k + 1 2 where k ≥ 2.…”

“…Let m be an integer and let q := e 2πiτ . Following [6], we also recall, the standard U(m) and V (m) operators by their action on q-series a(n)q n as follows,…”

On 𝑝-adic harmonic Maass functions

“…In many cases it is not hard to see that certain p-adic properties of q-series are propagated through the lifts. Some of these properties have been explored by Bringmann-Guerzhoy-Kane [6].…”

“…From Zagier's work and the work of Duke-Jenkins, it is not hard to see that certain p-adic properties of q-series are propagated through the lifts. Some of these properties have been explored by Bringmann-Guerzhoy-Kane [6]. In the case of weakly holomorphic modular forms, the CM values which make up the traces are algebraic numbers, making inherent p-adic properties much easier to explore.…”