Delta operation at height 2

TL;DR

This paper computes the delta power operation in Morava E-theory of height 2 at prime 3, providing explicit formulas and connecting it to elliptic curves, higher semi-additivity, and Hecke actions.

Contribution

It introduces an explicit formula for the delta power operation in Morava E-theory of height 2 at prime 3, linking it to elliptic curve moduli and Hecke operators.

Findings

Delta operation of a polynomial is itself a polynomial

Explicit formulas for delta power operation are derived

Connections to Hecke actions on elliptic curve functions are established

Abstract

We compute the delta power operation for morava E-theory of height 2 at the prime 3. The delta power operation was defined using the notion of higher semi additivity by Shachar Carmeli, Tomer M. Schlank and Lior Yanovski. We briefly survey the basic definitions in higher semi-additivity. Using explicit formulas for moduli spaces of elliptic curves and computations done by Y. Zhu for the total power operation we provide an explicit formula for the delta power operation. We obtain the numerical result that the delta operation of a polynomial is a polynomial which can be explained by a certain connection which was described by M. Hopkins, C. Rezk and later N. Stapleton between similar operations and the Hecke action on the functions on the moduli space of elliptic curves.