Unramified torsors

Abstract: We introduce unramified torsors as a tool for further studying local-global questions. A class $\tau \in H^1(k,G)$ is unramified at a place v if it extends over $\mathcal O_{k,v}$, i.e. if it lies in the image of $H^1(\mathcal{O}_{k,v},\mathcal G)\to H^1(k,G).$ For a finite set of places $S$ and an exact sequence $$1\to \mathcal G^0 \to \mathcal G \to \mathcal F \to 1$$ with $\mathcal{G}^0$ having connected fibers and $\mathcal{F}$ finite étale, we prove that the maps $$H^1_S(k,\mathcal G) \to H^1_S(k,\mathcal F) \to \prod_{v\in S} H^1(k_v,F)$$ have finite fibers, and that $H^{1}_S(k,\mathcal{G})$ is finite when $k$ is a number field. These finiteness results are key to proving the finiteness of Selmer sets, which, e.g., offers one route to weak Mordell-Weil in the case of abelian varieties. Along the way, we analyze torsors over finite fields using Lang’s theorem.

Reference: Poonen, Rational points on varieties, 6.5.7.

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.