Class 
Chapter 
Topics 
Exercises 
Suggested readings 
Jan 23 
I) Galois representations. 1) Galois groups 
Krull Topology of Galois group. Structure of the Galois groups of finite, local and number fields)
 

Jan 27 
I)1) (continued) 2) Galois representations 
1) Galois groups G_{K,S}. 2)Definition of a Galois representation. Unramified representation. Frobeniuses and Cebotarev. Galois representations over the complex numbers. 

Jan 30 
I)2) (continued) 3) Examples 
2) Galois representations over the complex numbers, over a finite field, over an ladic field.
3) Examples : The cyclotomic character. 

Feb 3 
I)3) Examples 
The cyclotomic character. The Tate representation attached to an elliptic curve. 
Exercices 1

Feb 7 
I)3) Examples 
The Tate representation attached to an elliptic curve over a finite field, a local field, a number field. The irreducibility theorem. 

Serre's book Abelian ladic representation, chapter IV 
Feb 10 
I)3) Examples, II) Etale cohomology 
The irreducibility theorem. II) Etale cohomology 
Exercices 2

Feb 13 
II) Etale cohomology 
First properties of etale cohomology. How to compute cohomology of varieties over various fields 

Feb 24 
II) Etale cohomology 
The Lefchetz fixed point formula in topology and in algebaric geometry. The Weil conjecture (= Deligne's theorem). Applications 
Exercises 3

Hartshorne, Appendix C. 
Feb 27 
II) Etale cohomology III) Algebraic Hecke characters 
Summary of the properties of the Galois representations appearing in etale cohomology. Notions of an algebraic representation, of weight of a representation, of compatible systems of representation. III) Hecke characters 

March 3 
III) Algebraic Hecke characters 
III) Hecke characters. Hecke Characters of Artin type. The adelic norm. Hecke character attached a CM elliptic curve. Algebraic Hecke chracter 

March 6 
III) Algebraic Hecke characters. IV) Local study of Galois representations (the case $l=p$) 
III) Realizations of algebraic Hecke characters. IV) Tamely and wildly ramified extensions.

Exercises 4

March 10 (double course : 10:401pm) 
III) Algebraic Hecke characters. IV) Local study of Galois representations (the case $l \neq p$) 
III) Characterization of the realizations of algebraic Hecke characters. IV) Tamely and wildly ramified extensions.
 
SerreTate's Good reduction of Abelian varieties

March 17 
IV) Local study of Galois representations (the case $l\neq p$) 
IV). Sturcture of the tame inertia subgroup.
Grothendieck's monodromy theorem.

March 20 
IV) Local study of Galois representations (the case $l\neq p$)
V) Fontaine's theory 
IV). The case of an elliptic curve. V) Overview of Fontaine's theory.
 Exercises 5

March 24 
V) Local study of Galois representations
(the case $l=p$ or Fontaine's theory) 
V) Formalism of Fontaine's rings. 

March 27 
V) Local study of Galois representations
(the case $l=p$ or Fontaine's theory) 
V) Formalism of Fontaine's rings. Semilinear representations. Naive examples of Fontaine's rings. The case of $C_p$
  For a proof of Tate's theorem on invariants, see this paper of Sen (from a computer on campus). 
March 31 
V) Local study of Galois representations
(the case $l=p$ or Fontaine's theory) 
V) Non Abelian cohomology.  
For more, see Serre, Cohomologie Galoisienne, section 1.5 
April 3 
V) Local study of Galois representations
(the case $l=p$ or Fontaine's theory) 
V)The first interesting Fontaine's ring : C_p. Proof of Serre's proposition. Sen's theory

Exercises 6  For the proof of the cohomological computations admitted in class,
see this other paper of Sen

April 7 
V) Local study of Galois representations
(the case $l=p$ or Fontaine's theory) 
V) Sen's theory. The ring B_{HT} and HodgeTate representations. Faltings theorem.

