Purely additive reduction of abelian varieties with torsion

TL;DR

This paper investigates the reduction behavior of abelian varieties over local fields with torsion points, focusing on the case of elliptic curves and their possible Kodaira types.

Contribution

It provides a detailed analysis of the reduction types of abelian varieties with torsion points, especially classifying Kodaira types for elliptic curves.

Findings

Classification of reduction types for abelian varieties with torsion

Explicit description of Kodaira types for elliptic curves

Insights into reduction behavior over mixed characteristic fields

Abstract

Let $\mathcal{O}_K$ be a discrete valuation ring with fraction field $K$ of characteristic $0$ and algebraically closed residue field $k$ of characteristic $p > 0$. Let $A/K$ be an abelian variety of dimension $g$ with a $K$-rational point of order $p$. In this article, we are interested in the reduction properties that $A/K$ can have. After discussing the general case, we specialize to $g=1$, and we study the possible Kodaira types that can occur.