For a family $A$ in $\{0,...,k\}^n$, its deletion shadow is the set obtained from $A$ by deleting from any of its vectors one coordinate. Given the size of $A$, how should we choose $A$ to minimise its deletion shadow? And what happens if instead we may delete only a coordinate that is zero? We discuss these problems, and give an exact solution to the second problem.

In 2013, Bhargava-Shankar proved that (in a suitable sense) the average rank of elliptic curves over Q is bounded above by 1.5, a landmark result which earned Bhargava the Fields medal. Later Bhargava-Gross proved similar results for hyperelliptic curves, and Poonen-Stoll deduced that most hyperelliptic curves of genus g>1 have very few rational points. The goal of my talk is to explain how simple curve singularities and simple Lie algebras come into the picture, via a modified Grothendieck-Brieskorn correspondence.

Moreover, I’ll explain how this viewpoint leads to new results on the arithmetic of curves in families, specifically for certain families of non-hyperelliptic genus 3 curves.

We will provide a general overview on some recent work on non-archimedean parametrizations and their applications. We will start by presenting our work with Cluckers and Comte on the existence of good Yomdin-Gromov parametrizations in the non-archimedean context and a $p$-adic Pila-Wilkie theorem.   We will then explain how this is used in our work with Chambert-Loir to prove bialgebraicity results in products of Mumford curves.