Arithmetic Geometry and Number Theory RTG Seminar: Schur indices and the p-adic local Langlands program

Seminar | October 9 | 3:10-5 p.m. | 891 Evans Hall

 David Sherman, Stanford University

 Department of Mathematics

Let $G$ be a finite group and Π an irreducible complex representation of $G$ whose character is valued in the number field $k$. There is a canonical class in the Brauer group of $k$ that describes the obstruction to descending $G$ to a representation on a $k$-vector space. The question of determining this class (or its order in the Brauer group, known as the Schur index of Π) is very classical, dating to the turn of the 20th century. I will describe a new approach to computing the local Brauer obstructions at places above p (for special cases of $G$), based on using Colmez's Montreal functor (for $\mathrm {GL}_2(\mathbf Q_p)$) to transport the problem to an equivalent one about Galois representations. This also motivates a generalization of the Montreal functor to the setting of $\mathrm {SL}_2(\mathbf Q_p)$, which I will also discuss.

Seminar Format: The seminar consists of two 50-minute talks, a pre-talk (3:10-4:00) and an advanced talk (4:10-5:00), with a 10-minute break (4:00-4:10) between them. The advanced talk is a regular formal presentation about recent research results to general audiences in arithmetic geometry and number theory; the pre-talk (3:10-4:00) is to introduce some prerequisites or background for the advanced talk to audiences consisting of graduate students.