On the structure of elliptic curves over finite extensions of $\mathbb{Q}_p$ with additive reduction

TL;DR

This paper investigates the topological group structure of elliptic curves with additive reduction over finite extensions of p-adic fields, providing methods to determine this structure from defining coefficients in unramified cases.

Contribution

It offers a novel approach to deducing the topological group structure of elliptic curves over p-adic fields from their Weierstrass coefficients, especially in unramified extensions.

Findings

Topological group structure can be explicitly determined from Weierstrass coefficients.

The structure is characterized for elliptic curves with additive reduction over unramified extensions.

Provides a new perspective on the relationship between algebraic and topological properties of elliptic curves.

Abstract

Let $p$ be a prime and let $K$ be a finite extension of $\mathbb{Q}_p$. Let $E/K$ be an elliptic curve with additive reduction. In this paper, we study the topological group structure of the set of points of good reduction of $E(K)$. In particular, if $K/\mathbb{Q}_p$ is unramified, we show how one can read off the topological group structure from the Weierstrass coefficients defining $E$.