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.