American Institute of Mathematics, Palo Alto, California
Alexander Polishchuk, Alexei Skorobogatov, and Yuri Zarhin
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:
The workshop schedule.
A report on the workshop activities.