Geometric Euler systems and algebraic cycles

Tom Weston (University of Michigan)

The Selmer group of a Galois representation is a certain Galois
cohomology group which is conjeturally related to special values
of L-functions, generalizing the usual Selmer group of an abelian
variety and the conjecture of Birch and Swinnerton-Dyer. I will
define geometric Euler systems and explain how they yield
annihilators for Selmer groups. I will attempt to explain the
differences with the arithmetic approach of Kolyvagin, Rubin and
others; the geometric theory is vastly simpler but also somewhat
less effective. I will then explain how one can use algebraic
cycles to produce geometric Euler systems in the cohomology of
algebraic varieties. I will conclude with some examples from
the theory of modular forms.