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