Day & Time: Tuesday and Friday 10:40am — 12:00
Location: Goldsmith 0226
Instructor: Joel Bellaiche
Prerequisite, contents, bibliography
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. |