Finite descent obstruction on curves and modularity

Abstract: We review some facts about modular curves and Galois representations. Then, following Helm and Voloch, we show that the only obstruction to the existence of integral points on (integral models of twists of) modular curves is that of finite étale descent. This is proved using an existence theorem for elliptic curves over $\mathbb{Q}$ with some prescribed local Galois data.

Reference: Helm, Voloch, Finite descent obstruction on curves and modularity.

algebraic geometrynumber theory

Audience: advanced learners

( slides )

---

STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Spring 2026 topic: The descent obstruction.

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.

---