Sheffield Number Theory Group Reading Seminar

Galois Representations

Spring 2015


This term we will study Galois representations following mainly the lecture notes of Frazer Jarvis's graduate course.

The meetings will take place in room F-35 Hicks Building on Thursdays at 2pm.

Talks:

Title: Galois Theory and representations of profinite groups 
Speaker: Dan
Contents: Brief review of infinite Galois theory and basic results related to representations of profinite groups. Main source is Section 1 and 2 of [J].

Title: Structure of G_Qp 
Speaker: Haluk
Contents: A discussion of extensions of Qp and their Galois groups, inlcuding the Weil group. Main source is Section 3 of [J]. 

Title:Representations of G_Qp 
Speakers: Jayanta + Tobias
Contents: Deligne-Groethendieck theorem, discussion of 1-dimensional representations and class field theory. The classification of 2-dimensional l-adic Galois representations and their reductions. Sections 5 and 6 of [J] should be summarized. Should be spread to two talks. 

Title: Elliptic curves and associated Galois representations 
Speaker: David
Contents: Goal is to discuss the mod l and l-adic representations coming from elliptic curves.

Title: Classical modular forms: Basics 
Speaker: Rudolph
Contents: Basics, including newforms theory.

Title: Classical modular forms: Associated Galois representations 
Speaker: Frazer
Contents: Discussion of results of Shimura and Deligne.

Title: Basics of automorphic representations
Speaker: Donggeon
Contents: Basic definitions, together with a discussion of the passage between classical newforms and cuspidal automorphic representations. May be spread to two talks, or skipped altogether depending on the audience's interests.

Title: Fontaine-Mazur Conjecture and Serre's Conjecture
Speaker:
Contents: Discussion of the statements

Title:An introduction to deformation theory of Galois representations
Speaker: 
Contents: Discussion of basics up to the representability of the deformation functor. Lectures 1 and 2 of [G] should be summarized. May be spread to two talks.

Materials:


[B1] G.Boeckle. Lecture notes on Galois representations. Link 

[B2] G.Boeckle. Lecture notes on deformation theory of Galois representations (more advanced than those of Gouvea's below). Link 

[DS] F.Diamond, J.Shurman: Chapter 9 of their book has an elementary account of Galois representations. 

[DDT] H.Darmon, F.Diamond and R.Taylor. Long expository article on the proof of Fermat's Last Theorem. Link 

[F0] J.-M. Fontaine, Y. Ouyang. Book on Fontaine's theory of p-adic representations of G_Qp. Link 

[G] F.Gouvea: Expository article on deformation theory in the proceedings "Arithmetic Algebraic Geometry". 

[J] F.Jarvis. Notes from his 2006 course on Galois representations. Link 

[M] B. Mazur: Expository article discussing Galois representations among others. Link 

[R] K.Ribet. Expository article on Galois representations and modular forms. Link 

[T] R.Taylor. Extended version of his ICM talk on Galois representations. Link 

[W] G.Wiese: Lecture notes on Galois representations. Link