# Galois representations (MATH 203b, Spring 2009)

Day & Time: Tuesday and Friday 10:40am — 12:00

Location: Goldsmith 0226

Instructor: Joel Bellaiche

## Schedule of Lectures

 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 l-adic 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 l-adic 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 : 10:40-1pm) 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. Serre-Tate'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. Semi-linear 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 Hodge-Tate representations. Faltings theorem.