The goal of this course is to understand Tate’s work [2] on the comparison between the p-adic etale cohomology and the de Rham cohomology (in degree 1) of a proper smooth variety over a local field (of char 0). This theory can be regarded as an initiation of the whole p-adic Hodge theory in arithmetic geometry, which plays an important role in modern number theory. We plan to fill in the preliminary materials and the details needed to understand the paper of Tate. These include: group schemes, p-divisible groups, Galois cohomology of a local field, and the Hodge-Tate decomposition. Further topics may include: Dieudonne theory, Fontaine’s period rings, p-adic integration, and/or their connections to p-adic differential equations.