Genus 2 curves with bad reduction at one odd prime

Official URL: https://dx.doi.org/10.1017/nmj.2023.35

The problem of classifying elliptic curves over Q with a given discriminant has received much attention. The analogous problem for genus 2 curves has only been tackled when the absolute discriminant is a power of 2. In this article, we classify genus 2 curves C defined over Q with at least two rational Weierstrass points and whose absolute discriminant is an odd prime. In fact, we show that such a curve C must be isomorphic to a specialization of one of finitely many 1-parameter families of genus 2 curves. In particular, we provide genus 2 analogues to Neumann-Setzer families of elliptic curves over the rationals.