Even Galois representations and the cohomology of $\mathrm{GL}(2,\mathbb Z)$

Joint IAS/Princeton University Number Theory Seminar
Topic:Even Galois representations and the cohomology of $\mathrm{GL}(2,\mathbb Z)$
Speaker:Avner Ash
Affiliation:Boston College
Date:Thursday, April 20
Time/Room:4:30pm - 5:30pm/Fine 214, Princeton University

Let $F$ be a field of characteristic not equal to 2. Let $r$ be a 2-dimensional even Galois representation which is induced from an $F$-valued character of odd order of the absolute Galois group of a real quadratic field $K$. After imposing some additional conditions on the inducing character $c$, we attach $r$ to a Hecke eigenclass in the cohomology of $\mathrm{GL}(2,\mathbb Z)$ with coefficients in a certain countably-infinite-dimensional vector space over $F$. We hope that this work will enable us to prove some cases of a Serre-type conjecture for certain reducible 3-dimensional Galois representations. This is joint work with Darrin Doud.