On the Kodaira types of elliptic curves with potentially good supersingular reduction

TL;DR

This paper investigates how wild ramification in minimal extensions affects the Kodaira types of elliptic curves with potentially good supersingular reduction over Henselian discrete valuation domains.

Contribution

It provides new insights into restrictions on reduction types of elliptic curves under wild ramification and analyzes the impact of isogenies on these types.

Findings

Restrictions on Kodaira types under wild ramification of degree 2

Characterization of reduction types for potentially good supersingular elliptic curves

Analysis of reduction types of isogenous elliptic curves with these properties

Abstract

Let $\mathcal{O}_K$ be a Henselian discrete valuation domain with field of fractions $K$. Assume that $\mathcal{O}_K$ has algebraically closed residue field $k$. Let $E/K$ be an elliptic curve with additive reduction. The semi-stable reduction theorem asserts that there exists a minimal extension $L/K$ such that the base change $E_L/L$ has semi-stable reduction. It is natural to wonder whether specific properties of the semi-stable reduction and of the extension $L/K$ impose restrictions on what types of Kodaira type the special fiber of $E/K$ may have. In this paper we study the restrictions imposed on the reduction type when the extension $L/K$ is wildly ramified of degree $2$, and the curve $E/K$ has potentially good supersingular reduction. We also analyze the possible reduction types of two isogenous elliptic curves with these properties.