The natural transformation Ξ from L–theory to the Tate cohomology of Z/2 acting on K–theory (constructed in [WW2] and [WWd]) commutes with external products. Corollary: The Tate cohomology of Z/2 acting on the K–theory of any ring with involution is a generalized Eilenberg–MacLane spectrum, and it is 4–periodic.

Part I. First steps in p-adic Hodge theory 4 1. Motivation 4 1.1. Tate modules 4 1.2. Galois lattices and Galois deformations 6 1.3. Aims of p-adic Hodge theory 7 1.4. Exercises 9 2. Hodge–Tate representations 10 2.1. Basic properties of CK 11 2.2. Theorems of Tate–Sen and Faltings 12 2.3. Hodge–Tate decomposition 15 2.4. Formalism of Hodge–Tate representations 17 2.5. Exercises 24 3. Étale φ-m...

Tate sequences are an important tool for tackling problems related to the (ill-understood) Galois structure of groups of S-units. The relatively recent Tate sequence “for small S” of Ritter and Weiss allows one to use the sequence without assuming the vanishing of the S-class-group, a significant advance in the theory. Associated to Ritter and Weiss’s version of the sequence are connecting homo...

This will contain my notes on Iwasawa-Tate theory. This will be done first for number fields and simple algebras (a la Weil), giving some number theoretic results. The meat of these notes is a study of the Standard L-functions for GL(n). We extend this to arbitary simple algebras (a la Godement/Jacquet). This is a long term project. As of now, I have basically completed the first case. There ar...

For an elliptic curve E defined over a fieldK, the Tate-Shafarevich group X(E/K) encodes important arithmetic and geometric information. An important conjecture of Tate and Shafarevich states X(E/K) is always finite. Supporting this conjecture is a cohomological analogy between Mordell-Weil groups of elliptic curves and unit groups of number fields. In this note, we follow the analogy to constr...