Local invariants of isogenous elliptic curves

TL;DR

This paper studies how key invariants of elliptic curves, like discriminant and periods, change under prime degree isogenies, providing a near-complete classification over l-adic fields.

Contribution

It offers a comprehensive analysis of invariant transformations under isogenies, filling gaps in the classification over l-adic fields.

Findings

Classification of invariants under isogenies over l-adic fields

Summary table of invariant changes for prime degree p

Partial results on wild potentially supersingular reduction

Abstract

We investigate how various invariants of elliptic curves, such as the discriminant, Kodaira type, Tamagawa number and real and complex periods, change under an isogeny of prime degree p. For elliptic curves over l-adic fields, the classification is almost complete (the exception is wild potentially supersingular reduction when l=p), and is summarised in a table.