Classical and pseudo (Galois) representations (Spring 2007 - G6232)

Day & Time:MW 2:40pm-3.55pm

Location: 622 Math buildings

Examinations: An exam for this class will be organized on demand.

Prerequisites: A good familiarity with commutative algebra is needed as well as a knowledge of basics of representations theory (at least at the level of the book of J.-P. Serre, representations of finite groups, part 1 and 2). For the second part, it is also necessary to know the basics of Galois theory and algebraic number theory.

Objective: The aim of this course is to give an overview of the modern theory of finite dimensional representations and pseudo-representations (or pseudocharacters), their deformations, and their families, with an emphasis on the case of Galois representations and its applications to number theory.

Temptative Syllabus:

Part I : General theory

Week 1. Reminder of associative algebra : Idempotents, radical, Azumaya algebras, etc.

Week 2. Finite dimensional representations of a group or an algebra over a field. Rationality questions.

Week 3. Finite dimensional representations of a group over a commutative ring. The case of a discrete valuation ring. Stable lattices. Ribet's lemma and its generalization.

Week 4. Pseudocharacters. Definition, basic properties.

Week 5. Pseudocharacters. Proof of Taylor's and Rouquier-Nyssen theorems.

Week 6. Finer study of residually multiplicity free pseudo-characters.

Week 7. Simple applications of pseudocharacters : convergence of a sequence of representations; Mazur's theorem on existence of a deformation ring; pseudo-deformations; Taylor's theorem on Galois representations attached to Hilbert modular forms and similar results. etc...

Part II : Families of Galois representations

Week 8. Example of p-adic families of Galois representations.

Week 9 to 11. p-adic families of representations of p-adic Galois groups (or : "Fontaine's theory in family") : Sen's theory and weights, Kisin's theorem and its generalisation on continuation of crystalline periods, families of (\phi,Gamma)-modules and refinement, reducibility loci.

Week 12. p-adic families of global Galois representations.

Less-Temptative Syllabus

A) Basics on associative ring theory. (January)

1) Semi-simple rings. [1,2,8]

2) The Jacobson Radical. [1,2,8]

3) Central simple algebras, (Azumaya algebras - will be studied later), the Brauer group of a field. [1,3]

B) Finite dimensional representations over a field, and their characters.

1) When is a representation determined by its character ? (02/07)

2) Rationality questions.

C) Finite dimensional representations over a discrete valuation ring.

1) Generalities about Lattices

2) Building of $\Gl_n(K)$ and its fixed part by a representation (02/12 and 14) [10,11]

3) Ribet's lemma and its generalizations. (02/14,19) [10,12]

D) Pseudocharacters [7,9].

1) The Identity of Frobenius. Definition of a pseudocharacters. (02/26)

2) First properties of pseudocharcater. Kernel of a pseudocharacter, Cayley-Hamilton and faithfulness properties, idmepotents, radical.

3) Pseudo-characters over a separably closed field : a (slightly generalized) theorem of Taylor. (02/28)

4) Residually irreducible pseudo-characters over a strictly local henselian ring : a (slightly generalized) theorem of Rouquier and Nyssen. (done)

5) Residually multiplicty free pseudo-characters. Reducibility loci, Ribet's lemma in full generality. (done)

6) Applications to sequences of representations. (done)

E) Families of Galois representations (as pseudocharacters)

1) Deformation of Galois representations and pseudocharacters. The pseudocharacters proof of Mazur's theorem. Pseudodeformations in the reducible cases.

2) Eigenvarieties and construction of p-adic families of Galois representations.

3) Sen's theory.

4) Refined family. A generalization to pseudocharacters of a theorem of Kisin on Crystalline perod.

5) Reducibility loci of refined families.

6) Arithmetic applications.

Exercises

For exercises about part A, see the bibliography below, especially [1] and [2].

Set number 1 (about part B and C1 above)

Set number 2 (about part C2)

Set number 3 (about part C3)

Set number 4 (about part D1)

Set number 5 (about Azumaya Algebars and idempotents)