Isogenies of supersingular elliptic curves over finite fields and operations in elliptic cohomology

TL;DR

This paper explores the structure of stable operations in supersingular elliptic cohomology through isogenies of supersingular elliptic curves, providing new models and proofs related to the Morava change of rings theorem.

Contribution

It introduces a framework for explicit stable operations in elliptic cohomology using isogenies and morphisms, simplifying the proof of key theorems and connecting to Hecke algebra structures.

Findings

Provides a simple proof of the elliptic cohomology Morava change of rings theorem.

Models for stable operations expressed via isogenies and morphisms.

Links supersingular elliptic cohomology to Hecke algebra structures.

Abstract

We investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields. Our main results provide a framework in which we give a conceptually simple proof of an elliptic cohomology version of the Morava change of rings theorem and also gives models for explicit stable operations in terms of isogenies and morphisms in certain enlarged isogeny categories. We relate our work to that of G. Robert on the Hecke algebra structure of the ring of supersingular modular forms.