Hodge-Tate Theory

This thesis aims to expose the amazing sequence of ideas, concerning p-adic representations coming from geometry, that form the heart of what was called Hodge-Tate theory. This subject, initiated by Tate in the late ’60s in analogy to classical Hodge theory, leads in to the now vast and highly fruitful program of p-adic Hodge Theory. The central result of the theory is the Hodge-Tate decomposition for abelian varieties, which gives a comparison isomorphism relating the étale cohomology and the de Rham cohomology. The original proofs are by Tate’s seminal analysis of p-divisible groups ([Tat67]) in the case of good reduction and Raynaud’s generalization to bad reduction using the semistable reduction theorem ([Gro72]). Here we present two different approaches, with the goal of accessibility. The first, due to Fontaine, is an elegant proof in full generality without the deep algebraic geometry machinery from SGA 7. The second proof, by Coleman, is for the case of good reduction, and falls out of a more explicit analysis of the geometry of abelian varieties. To build up to this result, we will develop requisite theory on the Galois cohomology and ramification theory of p-adic fields. After proving the decomposition, we discuss how it fits into Fontaine’s general formalism of admissible representations and period rings, and undertake the construction of BdR and the theory of de Rham representations. Finally, we give examples of computing periods for elliptic curves.