SPEAKER: Arvind Nair, Tata Institute
TITLE:
On the motive of a Shimura variety
ABSTRACT:
Shimura varieties are higher-dimensional generalizations
of modular curves and their cohomology groups are a source of
interesting Galois representations which can be related to modular
forms. (In the modular curve case this is the relation between H^1
of the curve and weight 2 modular forms due to Eichler and Shimura.)
In this situation, one wants something more than a Galois representation,
namely a motive. (In the modular curve case this is roughly speaking the
Jacobian of the curve rather than simply its H^1.) I will explain what
this means and how to do this when the Shimura variety is noncompact (as
the most natural examples are), at least for the most interesting part
(i.e. the tempered part) of the cohomology.