Cohomological methods in abelian varieties

March 26 to March 30, 2012

at the

American Institute of Mathematics, Palo Alto, California

organized by

Alexander Polishchuk, Alexei Skorobogatov, and Yuri Zarhin

Original Announcement

This workshop will be devoted to the integral motive, Chow groups and etale cohomology of abelian varieties, and applications to arithmetic geometry.

Grothendieck's standard conjecture C asserts that for a smooth projective variety over a field, the Kunneth projectors with rational coefficients are classes of algebraic correspondences. This is known for abelian varieties. Moreover, one has a Kunneth decomposition of the motive of an abelian variety in the category of Chow motives with rational coefficients. A canonical and functorial decomposition was found by Deninger and Murre using a generalisation of the Fourier-Mukai transform on Chow groups introduced by Beauville.

Passing from rational to integral coefficients leads to many intriguing questions. Do there exist an integral Fourier-Mukai transform or an integral analogue of the Deninger-Murre decomposition? Do the divided powers exist in the even etale cohomology groups of an abelian variety? Does the Hochschild-Serre spectral sequence for the etale cohomology of an abelian variety with finite or integral coefficients degenerate?

These questions have applications to the computation of the Brauer group of abelian varieties and K3 surfaces. In the case when the ground field is an ''arithmetic'' field or a function field, understanding cohomology is crucial for studying the behaviour of the Mordell-Weil rank in elliptic pencils and in towers of function fields.

The main topics for the workshop are:

  1. Integral motive of an abelian variety. We propose to understand in detail the recent works on cohomology of abelian varieties with integral and finite coefficients, the integral Fourier transform, and the problem of divided powers in cohomology and Chow groups. The aim is to advance in some of the open questions mentions above.
  2. Brauer groups of abelian varieties and K3 surfaces. This theme builds on the previous one. The understanding of the Brauer group is crucial for the computation of the Brauer-Manin obstruction to the existence of rational points and weak approximation, an area from which many new conjectures emanate.
  3. Arithmetic applications. This theme focuses on the behaviour of the Mordell-Weil rank in the recent work of Mazur-Rubin (and its applications to Hilbert's Tenth Problem) and the work of Ulmer on rational points of abelian varieties in towers of function fields. Among possible applications are the unboundedness problem for the rank of abelian varieties, and the arithmetic of rational points on elliptic surfaces over number fields.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.