Examples of descent

Abstract: The main theorem of descent states that if $X$ is a smooth proper variety over global field $k$, $G$ is a smooth affine algebraic group over $k$, and $f : Z\to X$ is a $G$-torsor over $X$, then the $k$-rational points of $X$ correspond to a union of $k$-rational points of finitely many twists of $Z$. This reduces the problem of finding rational points of $X$ to finding rational points of finitely many twists of $Z$. We illustrate this theorem through several examples, including the Weak Mordell-Weil Theorem.

Reference: Poonen, Rational points on varieties, Section 8.3., 8.4.1-2, 8.4.4-5.

algebraic geometrynumber theory

Audience: advanced learners

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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.

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