Mod p modular forms and simple congruences

Abstract

In this article, we first give a complete description of the algebra of integer weight modular forms on the congruence subgroup $\Gamma _0(2)$ modulo a prime $p\ge 3$. This result parallels results of Swinnerton-Dyer in the $SL_2(\mathbb {Z})$ case, Katz on the subgroup $\Gamma (N)$ for $N\ge 3$, Gross on the subgroup $\Gamma _1(N)$ for $N\ge 4$ and Tupan on modular forms of half-integral weight on $\Gamma _1(4)$. Next, we use the theory of mod p modular forms on $\Gamma _0(2)$ to prove the non-existence of simple congruences for Fourier coefficients of quotients of certain integer weight Eisenstein series on $\Gamma _0(2)$. The non-existence of simple congruences for coefficients of quotients of Eisenstein series on $SL_2(\mathbb {Z})$ has been shown by Dewar.