#
The Tate conjecture

July 23 to July 27, 2007
at the

American Institute of Mathematics,
Palo Alto, California

organized by

Dinakar Ramakrishnan and Wayne Raskind

## Original Announcement

This workshop will be devoted to the conjecture of Tate which
characterizes the cohomology classes of algebraic cycles on an algebraic variety*X* over a field *k* that is finitely generated over the prime field in terms of
the fixed space of even-dimensional *l*-adic etale cohomology under the
action of the absolute Galois group *G* of *k*. Here *l* is a prime number
different from the characteristic of *k*. This conjecture is an arithmetic
analogue of the Hodge conjecture for varieties over the complex numbers.
While the Tate conjecture is over 40 years old and has been verified in
many cases, we still lack a true understanding of why it should hold in
general. The main purpose of the workshop is to bring experts in various
aspects of the field together to synthesize the known methods and then
develop strategies for understanding and attacking the conjecture in a
more general way.
The main topics on which we will focus include:

- many examples for which the Tate conjecture is known amount
to explicit computations of the rank of the group of algebraic
cycles of codimension
*i* modulo homological equivalence on *X*,
of the order of the pole of the corresponding L-function, and
of the dimension of the fixed space of the *l*-adic etale
cohomology group *H*^{2i}(X^{-};Q_{l}(i)) under the action of *G*; the conjecture
predicts that these three numbers are same. Here *X*^{-} denotes
scalar extension of *X* to a separable closure of *k*. One theme
of the conference will be to synthesize the methods used for
these computations and to suggest problems for students and
other young mathematicians in this direction.
- methods coming from the theory of motives that have been
applied recently by Andre and Milne. These have been especially
effective for varieties over finite fields. Another theme of the
conference will be to see how such methods may be applied in
other cases.
- the relation between the conjecture of Birch and Swinnerton-Dyer
for elliptic curves over function fields over finite fields and
the Tate conjecture for surfaces over finite fields. The idea here
is to see how methods used in this case might be transported
back to the number field case.
- Combining several methods and results, it appears as if the Tate
conjecture for divisors on Shimura varieties is within reach. We
will review these techniques and discuss what more needs to be
done to achieve this result.
- the method of Faltings using heights to prove the
Tate conjecture in some cases for abelian varieties and how this might be
adapted to other cases. The Tate conjecture grew out of such
a statement about abelian varieties and we could hope that a
proof of the Tate conjecture in more general cases might grow
out of Faltings' proof.

The goal is to find some "cross-fertilization" between these methods
that will increase the power of each of them. We expect to have a
proceedings that will summarize work done on the conjecture and suggest
some open problems that appear tractable.
This will be especially valuable for students.
The organizers hope that participants might be willing to share any
vague and/or unpublished ideas they might have about the
Tate conjecture in the working sessions, and that the atmosphere and format of
AIM workshops will be conducive to significant progress being achieved
during or shortly after the meeting.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.