The Tate conjecture

This workshop, sponsored by AIM and the NSF, will be devoted to the conjecture of Tate which characterizes the cohomology classes of algebraic cycles on an algebraic varietyX 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:

1. 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 H²ⁱ(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.
2. 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.
3. 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.
4. 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.
5. 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 will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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