A tour of the Betti, étale, de Rham, and crystalline cohomology

Abstract: This talk is a crash course on the properties of various cohomology theories that will be used in the Lawrence-Venkatesh proof. These include the Betti cohomology, étale cohomology, de Rham cohomology and crystalline cohomology. We review relevant structures on each of these cohomology groups of a smooth proper variety, state some comparison theorems, and explain how they come together to form the "big diagram" in the Lawrence-Venkatesh argument.

[Update] I uploaded my outline for the talk to (slides) below. It is only a skeleton of the talk instead of a complete write-up, so if you want to review the material, I recommend reading the first reference (it's only 2 pages), and if you would like more detail, look into the other references.

Reference:

$\bullet$ Poonen, $p$-adic approaches to rational and integral points on curves, Sections 5 and 6.

For more details:

$\bullet$ Deligne, Hodge cycles on abelian varieties, Section 1 (for Betti and de Rham cohomology).

$\bullet$ Lawrence and Venkatesh, Diophantine problems and $p$-adic period mappings, Section 3 (to see how cohomology theories are going to be used).

$\bullet$ Brinon and Conrad, CMI summer school notes on $p$-adic Hodge theory, Section 9.1 (for details on the ring $B_{cris}$ and the functor $D_{cris}$).

$\bullet$ Nicole, Cris is for Crystalline (notes for a seminar talk on crystalline cohomology) (for the definition and motivations for crystalline cohomology).

$\bullet$ Voisin, Hodge theory and complex algebraic geometry I, Chapter II (for de Rham cohomology and the Hodge decomposition).

algebraic geometrynumber theory

Audience: advanced learners

( slides )

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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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