at the

American Institute of Mathematics, San Jose, California

organized by

Dinakar Ramakrishnan and Wayne Raskind

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*under the action of^{2i}(X^{-};Q_{l}(i))*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The organizers hope that participants might be willing to share any vague and/or unpublished ideas they might have about the Tate conjecture in the working sessions, and that the atmosphere and format of AIM workshops will be conducive to significant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.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

A rank inequality for the Tate Conjecture over global function fields

by Christopher Lyons

Nonhomeomorphic conjugates of connected Shimura varieties

by James S. Milne and Junecue Suh

Split reductions of simple abelian varieties

by Jeff Achter