A canonical lift of Frobenius in Morava E-theory

TL;DR

This paper proves that the $p$th Hecke operator in Morava $E$-cohomology is congruent to the Frobenius mod $p$, extending known results from complex $K$-theory and providing tools for testing Rezk's criterion.

Contribution

It establishes a canonical lift of Frobenius in Morava $E$-theory via the $p$th Hecke operator, generalizing classical congruences.

Findings

$p$th Hecke operator is congruent to Frobenius mod $p$

The result generalizes the Adams operation congruence in $K$-theory

Provides a method to test Rezk's congruence criterion

Abstract

We prove that the $p$th Hecke operator on the Morava $E$-cohomology of a space is congruent to the Frobenius mod $p$. This is a generalization of the fact that the $p$th Adams operation on the complex $K$-theory of a space is congruent to the Frobenius mod $p$. The proof implies that the $p$th Hecke operator may be used to test Rezk's congruence criterion.