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

TL;DR

This paper employs homotopy theoretic methods and $ ext{E}_ ext{infty}$ complex orientations to establish $p$-adic Kummer congruences among derivatives of the Weierstrass elliptic function, linking number theory and algebraic topology.

Contribution

It introduces a novel application of $ ext{E}_ ext{infty}$-orientation techniques to derive $p$-adic congruences for elliptic functions, reversing the usual application of the Ando-Hopkins-Rezk machinery.

Findings

Proves $p$-adic Kummer congruences for elliptic derivatives

Utilizes homotopy theoretic methods in number theory

Connects $ ext{E}_ ext{infty}$-orientations with elliptic function congruences

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 K\"ummer 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.