The Demyanenko-Manin method and Mumford's inequality

Abstract: In this talk, we will discuss two theorems regarding the number of rational points on curves of genus $g \geq 2:$ the Demyanenko-Manin theorem and Mumford's inequality. We will begin with the Demyanenko-Manin theorem, which tells us how the existence of enough functions $f_i: C \rightarrow A,$ for some abelian variety $A,$ allows us to show the number of rational points on $C$ is finite. After outlining the proof of this theorem and discussing an application to modular curves, we will then sketch a proof of Mumford's inequality, which gives an asymptotic bound on the number of points of bounded height without knowing Falting's 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.