Computing bad reduction for genus 3 curves with complex multiplication
Goren and Lauter studied genus 2 curves whose Jacobians are absolutely simple and have complex multiplication (CM) by the ring of integers of a quartic CM-field, and showed that if such a curve has bad reduction to characteristic p then there is a solution to a certain embedding problem. An analogous formulation of the embedding problem for genus 3 does not suffice for explicitly computing all primes of bad reduction. We introduce a new problem called the Isogenous Embedding Problem (IEP), which we relate to the existence of primes of bad reduction. We propose an algorithm which computes effective solutions for this problem and exhibits a list of primes of bad reduction for genus 3 curves with CM. We ran this algorithm through different families of curves and were able to prove the reduction type of some particular curves at certain primes that were open cases in the literature. This is joint work with Elisa Lorenzo Garcia, Kristin Lauter, Pinar Kilicer, Adelina Manzateanu and Christelle Vincent.