A bound on the primes of bad reduction for CM curves of genus 3

TL;DR

This paper establishes bounds on primes of bad reduction for genus 3 curves with primitive CM, extending known results from genus 2 and providing implications for constructing curves over finite fields.

Contribution

It provides new bounds on primes of bad reduction for genus 3 CM curves, generalizing previous work on genus 2 and impacting algorithmic curve construction.

Findings

Bounds on primes of bad reduction for genus 3 CM curves.

Implications for denominators of invariants and class polynomials.

Extension of previous bounds from genus 2 to genus 3.

Abstract

We give bounds on the primes of geometric bad reduction for curves of genus three of primitive CM type in terms of the CM orders. In the case of genus one, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least two, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus two. In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators of invariants and class polynomials, which are important for algorithmic construction of curves with given characteristic polynomials over finite fields.