Group actions on curves and applicationsMPIM

by Alexandros Konstantinou (MPIM)

Tuesday Aug 25, 2026, 2:00 PM → 3:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

PLeaSANT

In this talk we will focus on one arithmetic setting: the action of a finite group G on a curve X. The idea is to pass to its Jacobian variety and let the representation theory of G do the work. This splits the Jacobian, up to isogeny, into smaller factors. We will explain this method and give some examples.

As an application, I will turn to Tate--Shafarevich groups. These play an important role in the study of rational points on abelian varieties, but they are notoriously difficult to compute, or even to prove finite. For an elliptic curve, the Tate--Shafarevich group has square order when finite. This is not always true for abelian varieties of higher dimension, a fact that was long overlooked and often misstated in the literature. We will show that every square-free positive integer appears as the square-free part of the order of the Tate--Shafarevich group of an abelian variety over the rationals.