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

2025/06/09 by Kiran Luecke, Eric D. Peterson, Luecke, Kiran +1
Computer Science · Mathematics · #11F33 #11F50 #55N22 #55S25 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2506.07420

openalex publication_date 2025/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use homotopy theoretic methods to prove congruence relations of number theoretic interest. Specifically, we use the theory of \mathbb E_∞ 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_∞-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_∞-orientations of Tate fixed-point objects.

Citations

• The Chromatic Nullstellensatz
• Algebraic theories of power operations
• Topological cyclic homology
• Strictly commutative complex orientation theory
• Topological modular forms with level structure

Related

• Kummer congruences arising from the mirror symmetry of an elliptic curve
• Congruence Relations Connecting Tate-Shafarevich Groups with Bernoulli-Hurwitz Numbers by Elliptic Gauss Sums in Eisenstein Case
• On the universal p-adic sigma and Weierstrass zeta functions
• On the transfer congruence between p-adic Hecke L-functions
• Congruences between Heegner points and quadratic twists of elliptic curves