Speaker: Elia Gorokhovsky (Harvard)

Title: 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.