Bad reduction of genus $3$ curves with complex multiplication

TL;DR

This paper establishes bounds on primes where genus 3 curves with complex multiplication exhibit bad reduction, specifically when their stable reduction contains three genus 1 components, under certain CM-field conditions.

Contribution

It provides new bounds on primes for the bad reduction of genus 3 curves with complex multiplication by sextic CM-fields without imaginary quadratic subfields.

Findings

Bound on primes with specific bad reduction behavior

Conditions on the CM-field influence reduction properties

Results applicable to genus 3 curves with complex multiplication

Abstract

Let $C$ be a smooth, absolutely irreducible genus-$3$ curve over a number field $M$. Suppose that the Jacobian of $C$ has complex multiplication by a sextic CM-field $K$. Suppose further that $K$ contains no imaginary quadratic subfield. We give a bound on the primes $\mathfrak{p}$ of $M$ such that the stable reduction of $C$ at $\mathfrak{p}$ contains three irreducible components of genus $1$.