Number theory learning seminar 2015-2016

The seminar will meet Wednesdays 1:30--3:30pm in Room 384H. This year's seminar will focus on the BSD Conjecture, beginning with Tate's classic work, and its vast generalization by Bloch and Kato. Familiarity with abelian varieties and various other topics in arithmetic geometry (schemes, etale cohomology, class field theory, etc.), or a willingness to take on faith such machinery, will be assumed whenever needed to get through a lecture in finite time.

In the fall we will largely focus on Tate's ideas (both through global Galois cohomology and the relationship to surfaces in the function field case; i.e., the Artin-Tate conjecture) and address some work of others which built on that. In the winter and spring we will discuss the work of Bloch and Kato (and its precursors by Deligne and Beilinson), which rests on input from K-theory, p-adic Hodge theory, and motivic cohomology (each to be explained in a form that can be used for our purposes).

Here are some references relevant to this year's seminar:
Notes on elliptic curves II, Birch, Swinnerton-Dyer
Chow trace and Lang-Neron theorem, Conrad
Duality theorems in Galois cohomology over number fields, Tate
Arithmetic duality theorems, Milne
Notes on etale cohomology of number fields, Mazur
Etale cohomology and duality in number fields, Zink
Minimal models for elliptic curves, Conrad
Le Groupe de Brauer III, Grothendieck
Linking Artin-Tate and BSD, Gordon
On the conjectures of Birch and Swinnerton-Dyer, Tate
A note on height pairings, Tamagawa numbers, and the BSD Conjecture Bloch
On a conjecture of Artin and Tate, Milne
Curves and Jacobians over function fields, Ulmer
Cassels-Tate pairing on polarized abelian varieties, Poonen-Stoll
Neron models, Lie algebras, and reduction of curves of genus one, Liu-Lorenzini-Raynaud
On the Brauer group of a surface, Liu-Lorenzini-Raynaud
Values of L-functions and periods of integrals, Deligne
Higher regulators and values of L-functions, Beilinson
Introduction to the Beilinson conjectures, Schneider
Beilinson's conjectures, Nekovar
K_2 and L-functions of elliptic curves, Bloch-Grayson
An introduction to the conjecture of Bloch and Kato, Bellaiche
L-functions and Tamagawa numbers of motives, Bloch-Kato
The equivariant Tamagawa number conjecture: a survey, Flach



Notes -- use at your own risk.

These are informal notes. They may change without warning.

Fall quarter
1 Sept.30 Conrad/Venkatesh Overview .pdf
2 Oct. 7 Conrad/Venkatesh Bloch-Kato Conjecture and height pairings .pdf
3 Oct. 14 Conrad Neron models, Tamagawa factors, and Tate-Shafarevich groups .pdf
4 Oct. 21 Masullo Global duality in Galois cohomology .pdf
5 Oct 28 Venkatesh BSD Examples .pdf
6 Nov. 4 Booher Isogeny invariance over number fields .pdf
7 Nov. 11 Sherman Cassels-Tate pairing and relation to polarizations (Poonen-Stoll) .pdf
8 Nov. 18 Conrad Artin-Tate, fibered surfaces, minimal regular proper model .pdf
9 Dec. 2 Greer Brauer group and relation to Tate-Shafarevich group .pdf
10 Dec. 9 Rosengarten Tate's results via Artin-Tate: rank inequality, relation with BSD .pdf
Winter quarter
11 Jan. 6 Yun Equivalence of Artin-Tate and BSD over global function fields .pdf
12 Jan. 13 Lawrence Deligne's conjecture for critical L-values I .pdf
13 Jan. 20 Lawrence Deligne's conjecture for critical L-values II .pdf
14 Jan. 27 Feng The Bloch-Kato Selmer group .pdf
15 Feb. 3 Feng Bloch-Kato baby version: order of vanishing .pdf
16 Feb. 10 Silliman Heights .pdf
17 Feb. 17 Venkatesh Beilinson's conjectures I .pdf
18 Feb. 24 Venkatesh Beilinson's conjectures II .pdf
Spring quarter
19 3/30-4/6 Pollack Beilinson's conjectures III: examples .pdf
20 4/13-4/20 Silliman Statement of Bloch-Kato Conjectures and Comparison with BSD .pdf
21 April 27 Tony Mazur--Tate Conjecture .pdf
22 May 4 Venkatesh Beilinson's Conjecture for number fields .pdf