The Kodaira--Parshin family

Abstract: In this talk, we describe the construction of an abelian-by-finite family parametrized by a prime $q$ which is amenable to direct monodromy computations. This family, the Kodaira-Parshin family, serves as input to the arguments in Section 6 which use an abelian-by-finite family with full monodromy to prove finiteness of rational points. The bulk of the talk focuses on the ``finite'' part of abelian-by-finite: we will describe the construction of an \'etale cover of a curve $Y$ parametrizing $G$-covers of $Y$ branched at a single point, together with a universal curve.

Reference:

$\bullet$ Lawrence and Venkatesh, Diophantine problems and $p$-adic period mappings, Section 7.

algebraic geometrynumber theory

Audience: advanced learners

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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.

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