# \$p\$-adic Limit of Weakly Holomorphic Modular Forms of Half Integral Weight - Mathematics Number Theory

Abstract: Serre obtained the p-adic limit of the integral Fourier coefficient ofmodular forms on \$SL 2\mathbb{Z}\$ for \$p=2,3,5,7\$. In this paper, we extendthe result of Serre to weakly holomorphic modular forms of half integral weighton \$\Gamma {0}4N\$ for \$N=1,2,4\$. A proof is based on linear relations amongFourier coefficients of modular forms of half integral weight. As applicationswe obtain congruences of Borcherds exponents, congruences of quotient ofEisentein series and congruences of values of \$L\$-functions at a certain pointare also studied. Furthermore, the congruences of the Fourier coefficients ofSiegel modular forms on Maass Space are obtained using Ikeda lifting.

Author: Dohoon Choi, YoungJu Choie

Source: https://arxiv.org/