Siegel's method

Abstract: We will show how theorems about diophantine approximation (e.g., Roth's theorem that irrational algebraic numbers cannot be approximated too well by rational numbers) can be used to prove one of the most famous theorems of 20th century arithmetic geometry, Siegel's theorem that a hyperbolic affine curve can have only finitely many integral points. The proof is ineffective, however: 95 years later it is still not known if there is an algorithm that takes as input the equation of a curve and returns the list of its integral points.

Reference: Chapter 7 of Serre, Lectures on the Mordell-Weil theorem.

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.