Samuel Le Fourn

The problem of singular moduli admits several generalisations to higher-dimension, for example the following one. For a real quadratic field $F$ and $K/F$ a CM extension which is a quartic cyclic extension of $\mathbb{Q}$, consider the jacobians of curves of genus 2 with CM by $\mathcal{O}_K$ such that the curves themselves have potentially good reduction everywhere. Habegger and Pazuki proved that their height can be bounded (ineffectively) only in terms of $F$. In the present work (joint with Linda Frey and Elisa Lorenzo--Garcia), we provide (under GRH) an explicit bound in $F$. In this talk, I will explain some of the main difficulties to be overcome to obtain explicit bounds, and the somewhat surprising link with questions about Minkowski's bound that arise.