Rationality and p-adic properties of reduced forms of half-integral weight

TL;DR

This paper investigates the properties of special bases of half-integral weight modular forms, establishing criteria for Fourier coefficient integrality, and deriving congruences that extend known results for traces of singular moduli.

Contribution

It introduces a criterion for Fourier coefficient integrality and derives new congruences for these coefficients using recursive relations of Hecke operators.

Findings

Established integrality criteria for Fourier coefficients.

Derived new congruences extending known results.

Connected Fourier coefficient relations with traces of singular moduli.

Abstract

In this paper we study special bases of certain spaces of half-integral weight weakly holomorphic modular forms. We establish a criterion for the integrality of Fourier coefficients of such bases. By using recursive relations between Hecke operators, we derive relations of Fourier coefficients of each basis element and obtain congruences of the Fourier coefficients, which extend known congruences for traces of singular moduli.