p-adic congruences in iterated derivatives of the Weierstrass elliptic function

Abstract

We use homotopy theoretic methods to prove congruence relations of number theoretic interest. Specifically, we use the theory of ${\mathbb{E}}_{\infty }$ complex orientations to establish $p$-adic Kümmer congruences among iterated derivatives of the Weierstrass elliptic function. The machinery of Ando, Hopkins, and Rezk was developed with the intended application of taking congruence relations as input and producing ${\mathbb{E}}_{\infty }$-orientations as output. We run their machine in reverse, using as input the recent results of Carmeli and the first author on the existence of ${\mathbb{E}}_{\infty }$-orientations of Tate fixed-point objects.