The cohomology of the group Sp(2g, Z) of symplectic 2g x 2g integer matrices is known to

stabilize: in each given degree it reaches a "limit" as g goes to infinity. I will first discuss this stable

cohomology and how it arises naturally in topological problems. This cohomology

can also be interpreted as the cohomology of a moduli space of abelian varieties, and as such

(if taken with finite coefficients) it carries an action of the absolute Galois group of the rational numbers.

I will explain how to compute this action, and why I find the answer interesting.

This is all joint work with Tony Feng (Stanford) and Soren Galatius (Copenhagen).