Determining the primes of bad reduction of CM curves of genus 3

TL;DR

This paper introduces the Isogenous Embedding Problem (IEP) related to primes of bad reduction of genus 3 CM curves, providing an algorithm to determine reduction types and solving previously open cases.

Contribution

It defines the IEP, links it to bad reduction primes of genus 3 CM curves, and offers an algorithm to compute solutions, advancing understanding of reduction behavior.

Findings

Solved open cases of reduction types at specific primes

Developed an algorithm for the IEP

Linked IEP solutions to primes of bad reduction

Abstract

In this paper we introduce a new problem called the Isogenous Embedding Problem (IEP). The existence of solutions to this problem is related to the primes of bad reduction of CM curves of genus $3$ and we can detect potentially good reduction in absence of solutions. We propose an algorithm for computing the solutions to the IEP and run the algorithm through different families of curves. We were able to prove the reduction type of some particular curves at certain primes that were open cases in [LLLR21].