January 30, 2015

Workshop on The Equivariant Tamagawa Number Conjecture
to be held at McMaster University

September 27-28, 2003

Organizer: Manfred Kolster, McMaster University

Scientific Background

One of the most fascinating topics in Algebraic Number Theory and - more generally - in Arithmetic Algebraic Geometry is the arithmetic interpretation of special values at integer points of L-functions attached to varieties over a number field . Classical examples are provided by the analytic class number formula of Dirichlet, which describes the residue at 1 of the zeta-function of a number field in terms of the Dirichlet regulator, the class number and the order of the group of roots of unity, and by the Conjecture of Birch and Swinnerton-Dyer, which predicts the order of vanishing of the L-function of an elliptic curve E over the rationals at 1, and expresses the leading term via arithmetic data attached to E. A far-reaching generalization to L-functions of arbitrary smooth projective varieties (or motives ) over a number field is due to Bloch-Kato. They conjecturally described the leading terms of the values of the L-functions at integer points in terms of Tamagawa numbers. Using a reformulation of the Bloch-Kato Conjecture due to Fontaine and Perrin-Riou - and independently Kato -, Burns and Flach extended the conjecture to take into account the action of endomorphisms of the variety. This Equivariant Tamagawa Number Conjecture (ETNC) encompasses the refinements of various classical conjectures, e.g. Gross' refinement of the Birch and Swinnerton-Dyer Conjecture for CM elliptic curves, and all the conjectures of Chinburg and others in Galois module theory.

In the special case of the Dedekind zeta-function of a number field the original Bloch-Kato Conjecture is equivalent to a cohomological version of a conjecture of Lichtenbaum, which expresses the leading term of the zeta-function at negative integers as a non-zero rational multiple of the Borel regulator, where the rational number is given as an Euler characteristic in etale cohomology. For abelian number fields this conjecture was proved in 1996 by Nguyen Quang Do, Kolster and Fleckinger up to powers of 2.
In the last 3 years there has been increased activity in this field with striking results: Benois and Nguyen Quang Do proved the full Bloch-Kato Conjecture for abelian number fields (up to powers of 2) by showing the compatability of the conjecture with the functional equation, Ritter and Weiss proved an equivariant version of the so-called Main Conjecture in Iwasawa-theory - a key ingredient in the study of the ETNC - for relative abelian extensions, which implied the validity of the ETNC for values at 0 for abelian fields, Huber and Kings gave a different approach to parts of these results using Euler systems, and finally Burns and Greither proved the ENTC for all abelian number fields - as always up to powers of 2.

Please note:

In preparation for the workshop D. Benois will give two introductory talks on the Equivariant Tamagawa Number Conjecture on Thursday Sept. 18 and Wednesday Sept. 24 at 15:30 in HH 312.
M. Flach will give a colloquium talk for a general audience "The Equivariant Tamagawa Number Conjecture" on Friday, Sept. 26, 15:30 in HH 217.

Tentative Schedule

All talks take place in Room 217 in Hamilton Hall (HH), the new location of the Department of Mathematics at McMaster University.
Saturday, Sept. 27
9:00 -- 10:00 M. Flach (CalTech):
"The Equivariant Tamagawa NumberConjecture - details of some proofs"
10:30 -- 11:30 D. Burns (King's College)
"On the leading terms and values of equivariant motivic L-functions".
12:00 -- 13:00 V. Snaith (U. Of Southampton)
"Annihilation of etale cohomology in the non-abelian case"
Lunch break  
15:00 -- 16:00 A. Weiss (U. of Edmonton)
"Some steps towards equivariant Iwasawa theory in the noncommutative situation, part I"
16:30 -- 17:30 . J. Ritter (U. of Augsburg)
"Some steps towards equivariant Iwasawa theory in the noncommutative situation, part II"
Sunday, Sept. 28
9:00 -- 10:00 D. Benois (U. of Bordeaux)
"On Tamagawa numbers of crystalline representations"
11:00 -- 12:00 S. Kim (McMaster U.)
"Non-abelian evidence for the Burns-Flach Conjecture in the number field case"
14:00 -- 15:00 T. Nguyen Quang Do (U. of Besancon)
"On the (p)-ramified Iwasawa module"

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