The Galois action on the stable homology of symplectic groups over Z.

Joint IAS/Princeton University Number Theory Seminar
Topic:The Galois action on the stable homology of symplectic groups over Z.
Speaker:Akshay Venkatesh
Affiliation:Stanford University; Distinguished Visiting Professor, School of Mathematics
Date:Thursday, February 8
Time/Room:4:30pm - 5:30pm/Fine 214, Princeton University

The Galois group of Q acts on the homology of the complex moduli space of abelian varieties, or, equivalently, on the homology of symplectic groups Sp_{2g}(Z). (Here we take homology with finite or profinite coefficients.) In particular, the Galois group acts on the stable homology. Perhaps surprisingly, this action does not factor through  powers of the cyclotomic character. Rather, one sees nontrivial extensions between the torsion part and the torsion-free quotient. I will explain how to see this, and some consequences for families of abelian varieties. This is joint work with Tony Feng and Soren Galatius.