A remark on Tate's algorithm and Kodaira types

TL;DR

This paper revisits Tate's algorithm for elliptic curves, providing a new characterization of Kodaira types based on valuations, and explores their behavior under tame extensions of local fields.

Contribution

It offers a reformulation of Tate's algorithm to identify Kodaira types via valuation minima and analyzes their behavior in tame local field extensions.

Findings

Characterization of Kodaira types using valuation minima

Behavior of Kodaira types in tame extensions analyzed

Reformulation of Tate's algorithm for minimal models

Abstract

We remark that Tate's algorithm to determine the minimal model of an elliptic curve can be stated in a way that characterises Kodaira types from the minimum of v(a_i)/i. As an application, we deduce the behaviour of Kodaira types in tame extensions of local fields.