Abstract:

We will discuss some of the amazing ideas regarding p-adic representations coming from geometry, initiated by Tate in the late ’60s in analogy to classical Hodge theory, and leading to the what we now call p-adic Hodge Theory.
The central result of the theory is the Hodge-Tate decomposition for abelian varieties, which gives a first \'etale-de Rham comparison isomorphism. We will mainly focus on Tate's analysis of p-divisible groups, but will first give a fairly broad amount of motivation for the study of p-divisible groups, also in sight of the next talk.
Time permitting, we will discuss the theory of period rings via the classification of vector bundles on the Fargues-Fontaine curve, which we would like to sketch.