Speaker: Alex Best, 14:00-14:20

Title: p-adic integration and points on curves

Abstract: The problem of algorithmically determining the set of rational and integral points on curves has seen impressive progress in the last 20 years, especially in the area surrounding Chabauty's method.
I'll give an overview of recent work involving $p$-adic integration applied to such problems and their strengths and limitations.

Speaker: Sebastian Monnet, 14:30-14:50

Title: S4-quartics with prescribed norms.

Abstract: Let $K$ be a number field with $\mathbb{Q}$-basis $\{e_1, ..., e_n\}$, and let $\alpha$ be a rational number. It is natural to ask whether the "norm equation" $N_{K/Q}(x_1e_1 + ... + x_ne_n) = \alpha$ has rational solutions. Since the answer depends only on $K$, we may ask how often this norm equation has rational points as we vary $K$. The case of abelian number fields was solved by Frei-Loughran-Newton, and in this talk we present one of the simplest non-abelian cases: $S_4$-quartics.