Pacific Institute for the Mathematical Sciences - PIMS

Title: On the motive of a Shimura variety
Speaker: Arvind Nair (Tata Institute for Fundamental Research, Mumbai, Maharashtra, India)
Date and Time: Monday, April 24 2006, 01:30 PM - 02:30 PM
Location: West Mall Annex Room 110 (Coffee at 1st floor PIMS lounge), PIMS-UBC
Abstract: Shimura varieties are of interest as a source of
motives/Galois representations about which much can be said using the
methods of representation theory. For various reasons one would like to
have a Grothendieck motive for such a variety. If the Shimura variety
is projective this is immediate. If it is not projective (as in most
classical examples, e.g. the moduli space of principally polarized
abelian varieties), the desired Grothendieck motive should realize to
the intersection cohomology of the minimal compactification of Baily
and Borel. I'll show that if the Shimura variety is related to abelian
varieties, then a motive can be constructed which realizes to the
subspace of intersection cohomology satisfying the generalized
Ramanujan conjecture at (any) one finite prime. (This is the
'essential' part according to standard conjectures in representation
theory.) In the case when the Shimura variety is a modular curve this
specializes to the Eichler-Shimura-Deligne-Scholl motive for classical
modular forms.

Title: The Hodge conjecture for the primitive cohomology of theta divisors
Speaker: Elham Izadi (University of Georgia in Athens, Georgia, USA)
Date and Time: Monday, April 24 2006, 03:00 PM - 04:00 PM
Location: West Mall Annex Room 110 (Coffee at 1st floor PIMS lounge), PIMS-UBC
Abstract: I will first discuss the meaning of the Hodge conjecture in
general and then specialize to abelian varieties. The primitive
cohomology of the theta divisor of an abelian variety gives a special
Hodge structure for which one can ask whether the Hodge conjecture is
true. Using Prym-embedded curves, this question was answered in the
affirmative by myself and van Straten for abelian fourfolds. In this
talk which is about joint work with Csilla Tamas, I will discuss the
case of abelian fivefolds and show in particular that Prym-embedded
curves do NOT solve the Hodge conjecture. I will, however, introduce a
different family of curves which is very likely to give an answer to
the Hodge conjecture.

Title: On the motive of the stack of bundles
Speaker: Kai Behrend (University of British Columbia)
Date and Time: Monday, April 24 2006, 04:30 PM - 05:30 PM
Location: West Mall Annex Room 110 (Coffee at 1st floor PIMS lounge), PIMS-UBC
Abstract: Let G be a split connected semisimple group over a field K.
We give a conjectural formula for the motive of the stack of G-bundles
over a curve C, in terms of special values of the motivic zeta function
of C. The formula is true if C=P1 or G=SLn. If K=C, upon applying the
PoincarŽ or Serre characteristic, the formula reduces to results of
Teleman and Atiyah-Bott on the gauge group. If K=Fq, upon applying the
counting measure, it reduces to the fact that the Tamagawa number of G
over the function field of C is |¹1(G)|. This is joint work with Ajneet
Dhillon.