Class field theory

Reading group, Michaelmas 2009

Meeting on Fridays at 2pm in T14 (third floor on the left) .

SCHEDULE

Talk 1: 23/10/2009 (Martin Mereb)

Group cohomology, restriction and inflation, long exact sequence, induction and inflation,

some examples, in particular Gal(L,K)-modules; [N I,1-4 or AW 1-3,6]), Cup-product, sketch

of proof of Tate's theorem ([M] II.3.11), periodicity of cohomology for cyclic groups....

Talk 2: 30/10/2009 (Frank Gounelas)

Galois extensions of local fields (main theorem of infinite Galois theory, complete

discrete valuation rings, classification of local fields, unramified extensions, units of

a local field, Galois group of maximal unramified extension of a local field; [N2

II.7.1-5])

Talk 3: 17/11/2009 (Tobias Barthel)

Galois cohomology and Brauer Groups (application to Galois group of extension L/K acting

on L*; [S X.5]), Brauer groups (Link to central simple algebras; [S2].1, [S] and [M], Severi-Brauer

varieties). Abelian extensions of local fields (application of Tate's theorem to abelian

extensions of local fields, explicit description for unramified extensions; [S2 2.1-6,8]).

Talk 4: 20/11/2009 (Tobias Barthel, Frank Gounelas)

Local class field theory: Computation of the Brauer group of the reals, finite fields.

Local invariants, computation of the Brauer group of a local field [S] [M], fundamental class [M],[N]

Talk 5: 27/11/2009 (Frank Gounelas)

Invariants, cup products, symbols and the local reciprocity map. Kummer/Artin-Schreier symbols [S]. Local

Existence theorem ([S XIV.6 and XI.5]), local Kronecker-Weber and maximal abelian extension of Q_p.

Talk 6: 4/12/2009 (Tobias Barthel)

Idele class groups, Hasse principle for central simple algebras, global reciprocity, fundamental exact

sequence for brauer groups (Hasse-Brauer-Noether theorem). [M], [N2], [N3]

Talk 7 : 9/12/2009 (Damiano Testa)

Proof of Chebotarev density, and other analytic bits and pieces for the proofs of the previous talk.

References:

[AW] Atiyah, Wall Cohomology of groups, in: Cassel, Froehlich: Algebraic number theory

[M] Milne's notes on class field theory

[N] Neukirch, Class field theory

[N2] Neukirch, Algebraic number theory

[N3] Neukirch, Cohomology of number fields

[S] Serre: Local fields GTM

[S2] Serre Local class field theory: Cassel, Froehlich: Algebraic number theory

[T] Tate Global class field theory: in Cassels and Froehlich