Reduction types of CM curves

TL;DR

This paper investigates the reduction behaviors of low genus curves with complex multiplication, classifying possible types for elliptic and genus 2 curves, and providing bounds on torsion subgroups over local fields.

Contribution

It offers a classification of reduction types for CM curves of genus 1 and 2, and establishes bounds on torsion subgroups for CM abelian varieties over local fields.

Findings

Classified Kodaira types for elliptic curves with CM

Determined possible Namikawa Ueno types for genus 2 CM curves

Provided bounds on torsion subgroups of CM abelian varieties

Abstract

We study the reduction properties of low genus curves whose Jacobian has complex multiplication. In the elliptic curve case, we classify the possible Kodaira types of reduction that can occur. Moreover, we investigate the possible Namikawa Ueno types that can occur for genus $2$ curves whose Jacobian has complex multiplication which is defined over the base field. We also produce bounds on the torsion subgroup of abelian varieties with complex multiplication defined over local fields.