The moments of the Weierstrass $P$-function can be computed recursively and are quasi-modular forms. They can be completed to almost-holomorphic modular forms via the Legendre period relation. I will explain how to do the same computation using calculus on the Tate curve model for the elliptic curve which is closely related to Schottky uniformization. By comparing the two approaches, this produces many interesting identities between polynomials of the Eisenstein series and series of Lambert type, which seem to be difficult to prove otherwise. I will also explain how the Weierstrass $P$-function and its moments are related to characters of certain Vertex algebras. If time permits, I will discuss the connection between the calculus on the Tate curve and the computation of correlation functions in certain conformal field theory on the torus, and mention the relation (via mirror symmetry) to the counting of Hurwitz numbers.

This is work in progress with Si Li.