Families of varieties with good reduction

Abstract: We review some constructions on crystalline representations and cohomology. Then we present a result in Lawrence-Venkatesh that shows the points in a residue disk that define semisimple representations are contained in a proper analytic subset. The proof illustrates the basic strategy in Lawrence-Venkatesh: to show the finiteness of a set of points, one only need to show its image in the period domain is contained in a Zariski-closed subset with dimension smaller than that of the orbit of a point under the complex monodromy group.

Reference:

$\bullet$ Lawrence and Venkatesh, Diophantine problems and $p$-adic period mappings, Section 3.

$\bullet$ Faltings, Crystalline cohomology and p-adic Galois-representations. Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 25–80.

$\bullet$ Illusie, Crystalline cohomology. Section 3.Motives (Seattle, WA, 1991), 43–70.

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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