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" |
| Coffeebreak |
|
| 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" |
| Coffeebreak |
|
| 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" |
| Coffeebreak |
|
| 11:00 -- 12:00 |
S. Kim (McMaster U.)
"Non-abelian evidence for the Burns-Flach Conjecture in the
number field case" |
| Lunchbreak |
|
| 14:00 -- 15:00 |
T. Nguyen Quang Do (U. of Besancon)
"On the (p)-ramified Iwasawa module" |
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