Symplectic K-theory of the integers and Galois groups.

Geometry Topology Seminar
Monday, December 4, 2017 - 14:00
1 hour (actually 50 minutes)
Skiles 006
Stanford University
The general linear groups GL_n(A) can be defined for any ring A, and Quillen's definition of K-theory of A takes these groups as its starting point.  If A is commutative, one may define symplectic K-theory in a very similar fashion, but starting with the symplectic groups Sp_{2n}(A), the subgroup of GL_{2n}(A) preserving a non-degenerate skew-symmetric bilinear form.  The result is a sequence of groups denoted KSp_i(A) for i = 0, 1, ....  For the ring of integers, there is an interesting action of the absolute Galois group of Q on the groups KSp_i(Z), arising from the moduli space of polarized abelian varieties.  In joint work with T. Feng and A. Venkatesh we study this action, which turns out to be an interesting extension between a trivial representation and a cyclotomic representation.