Speaker: Xinyu Fang (Harvard)

Title: A tour of the Betti, etale, 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, etale 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.

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.

$\bullet$ 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).