Jared Weinstein: Period maps in p-adic geometry, Lecture 1

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. (Lecture 1 of 4)