Universal Finite Subgroup of the Tate Curve

TL;DR

This paper classifies finite subgroups of the Tate curve and explicitly constructs its universal finite subgroup using isogenies and Tate K-theory, advancing the understanding of elliptic curve moduli.

Contribution

It provides the first explicit construction of the universal finite subgroup of the Tate curve, extending prior theoretical discussions.

Findings

Classification of finite subgroups of the Tate curve

Explicit construction of the universal finite subgroup

Application of Tate K-theory and isogenies

Abstract

In their book Katz and Mazur discuss the moduli problem of the subgroup-schemes of elliptic curves. We give the classification of the finite subgroups of the Tate curve before. Moreover, Katz and Mazur define the universal finite subgroup of an elliptic curve. In this paper we give an explicit construction of the universal finite subgroup of the Tate curve via isogenies and the stringy power operation of Tate K-theory.