Uniformity for rational points

Abstract: We discuss the proof of Proposition 8.1 in [DGH], which gives a uniform bound for the intersection of rational points $C(\overline\mathbb{Q})$ of a curve $C$ of large modular height in an abelian variety $A$ and a finite rank subgroup $\Gamma\subseteq A(\overline\mathbb{Q})$. The number of large points can be handled by a standard application of the Vojta and Mumford inequalities. The key of [DGH] is to bound the number of those small points using the so-called New Gap Principle.

We then deduce the uniform boundedness of rational/torsion points of curves in [DGH], i.e., their Theorems 1.1, 1.2, and 1.4, from the above Proposition 8.1 (for curves of large modular height) and some other classical results (taking care of curves of small modular height).

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 2024 topic: The contents of Serre, Lectures on the Mordell-Weil theorem

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.