Congruences and Reduction mod p
The study of modular forms modulo p was originated by Swinnerton-Dyer [Sw D], who determined the structure of the algebra of modular forms mod p. Serre then showed how one can extend this theory in many ways, and in particular obtained results concerning the congruence properties mod higher powers of p for the coefficients of the q-expansions of modular forms. After laying the basic foundations for the q-expansions, we reproduce Swinnerton-Dyer’s results, and then some of Serre’s basic results, referring to his more extensive papers for the continuation of the theory.
KeywordsPower Series Modular Form Preceding Theorem Integer Coefficient Power Series Ring
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