The Eisenstein series are classic examples of integer weight modular forms while the Dedekind Eta function is a classic example of a half-integer weight modular form. In this talk we’ll present identities for quotients of Eisenstein series by powers of the Eta function, arising from the action of the Hecke operator on spaces with no cusp forms. A consequence of these identities is a systematic congruence for the coefficients of these functions modulo prime powers. We will also discuss the moments of cranks and ranks, which are “partition statistics”: that is, they are arithmetic functions which have something to say about integer partitions. Curiously, the congruences for the Eisenstein-Eta quotients imply similar results for these partition statistics, which we will also discuss.

Talk by Clayton Williams.