過去の記録 ~12/11次回の予定今後の予定 12/12~

開催情報 火曜日 16:30~18:00 数理科学研究科棟(駒場) 126号室
担当者 小林俊行


17:00-18:45   数理科学研究科棟(駒場) 号室
Birgit Speh 氏 (Cornell University)
Introduction to the cohomology of discrete groups and modular symbols 1 (English)
[ 講演概要 ]
The course is an introduction to the cohomology of torsion free discrete subgroups $\Gamma \subset G $ of a semi simple group $G$. The discrete group $\Gamma$ acts freely on the symmetric space $X= G/K$ and we will always assume that $\Gamma \backslash G/K$ is compact or has finite volume. An example is a torsion free subgroup $\Gamma_n $ of finite index n in Sl(2,Z) acting on $Sl(2.R)/SO(2) \simeq {\mathcal H}=\{z=x+iy \in C| y >0 \}$ by fractional linear transformations. $\Gamma_n \backslash {\mathcal H}$ can be determined explicitly and it can be visualized as an area in the upper half plane glued at the boundary. It is easy to see that it has some nice compactifications.

The cohomology $H^*(\Gamma, C)$ of the group $\Gamma$ is equal to the deRham cohomology $H^*_{deRham}(\Gamma \backslash X, C)$ of the manifold $\Gamma\backslash X$. This cohomology is studied by proving that it is isomorphic to the $H^*(g,K,{\mathcal A}(\Gamma \backslash G))$. Here ${\mathcal A}(\Gamma \backslash G)$ of automorphic functions on $\Gamma \backslash G$. In the case $\Gamma_n \subset Sl(2,Z)$ the space ${\mathcal A}(\Gamma \backslash G)$ is the space of classical automorphic functions on the upper half plane containing holomorphic cusp form, Eisenstein series, Maass forms and it is often introduced in an introductory course in analytic number theory.

On the geometric side we will construct some of the cycles (modular symbols) in the homology $H_*(\Gamma\backslash X)$ which are dual to the cohomology classes we constructed. In our example $\Gamma_n\backslash Sl(2,R)/SO(2)$ these cycles correspond to geodesics and can easily be visualized.

In this course I will explain these results and show how to use them to prove vanishing and non vanishing theorem for $H^*_{deRham}(\Gamma \backslash X)$. I will state the results in full generality, but I will prove them only in the classical case: G=SL$(2,R)$ and the subgroup $\Gamma= \Gamma_n$ a congruence subgroup. Some familiarity with Lie groups and Lie algebras is only prerequisite for the course.