Abelian-by-finite families I

Abstract: In this talk, we will begin the proof that there are only finitely many rational points whose pre-images along the finite map have large Galois orbits, introduced in the previous talk. The proof of this statement requires two lemmas: a generic simplicity statement, and a finiteness statement if we consider only the rational points corresponding to a given simple Galois representation. We will begin by presenting the proof, assuming these two lemmas, before proving the finiteness lemma.

We will follow part of Lawrence and Venkatesh, Diophantine problems and $p$-adic period mappings, Section 6.

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.

Fall 2025 topic: Weil conjectures.

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.