Focus Period on Representations in Arithmetic

This is the final segment of the PIMS Focus Group on Representations in Arithmetic. This week will focus on  a series of lectures on p-adic Geometry related to Scholze’s work. Speakers include: Pierre Colmez, Wiesia Niziol, Kiran Kedlaya, Jared Weinstein

Pierre Colmez and Wiesia Niziol:  

P-adic etale cohomology of p-adic symmetric spaces

We will present different ways to compute the p-adic etale cohomology of layers of the Drinfeld tower, and give applications to the p-adic local Langlands correspondence". 

Jared Weinstein

Period maps in p-adic geometry

On a complex variety, you can integrate a differential form over a cycle to get a period. For instance, an elliptic curve has two periods, whose quotient gives an element of the upper half plane. There is a family of concepts (Hodge decomposition, variation of Hodge structures, Shimura varieties) arising from the study of periods on families of complex varieties. What if the complex variety is replaced with a rigid-analytic variety over a p-adic field? We will review work of Tate, Fontaine, Kedlaya-Liu, Scholze and others that falls under the domain of p-adic Hodge theory. One goal will be to understand the surprising Hodge-Tate period map, defined by Scholze, attached to the modular curve at infinite level.

Kiran Kedlaya:

Multivariate (phi, Gamma)-modules

The classical theory of (phi, Gamma)-modules relates continuous p-adic representations of the Galois group of a p-adic field with modules over a certain mildly noncommutative ring. That ring admits a description in terms of a group algebra over Z_p which is crucial for Colmez's p-adic local Langlands correspondence for GL_2(Q_p). We describe a method for applying a key property of perfectoid spaces, the analytic analogue of Drinfeld's lemma, to the construction of "multivariate (phi, Gamma)-modules" corresponding to p-adic Galois representations in more exotic ways. Based on joint work with Annie Carter and Gergely Zabradi.  

These lectures are available via videoconference through the links below.