The descent obstruction

Abstract: In this talk, we will briefly review the notion of Selmer sets and how they provide an obstruction to the existence of rational points. We will then prove a theorem showing that, the Selmer set is finite in interested cases. Next, we will define the descent obstruction to the local-global principle and compare it with the Brauer-Manin obstruction. Finally, we will construct explicit torsors over Iskovskikh's surface to demonstrate that it exhibits a descent obstruction and, in particular, possesses no rational points. We may also apply descent obstruction to see the failure of strong approximation if time permits.

Reference: Poonen, Rational points on varieties, Section 8.1,8.4, 8.5.1.

algebraic geometrynumber theory

Audience: advanced learners

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.