Lefschetz numbers of symplectic involutions on arithmetic groups

TL;DR

This paper derives a precise formula for Lefschetz numbers of symplectic involutions acting on the cohomology of arithmetic groups related to central simple algebras, connecting algebraic and geometric properties.

Contribution

It provides a new explicit formula for Lefschetz numbers of symplectic involutions on arithmetic groups, using smoothness properties of associated group schemes.

Findings

Derived a formula for Lefschetz numbers of symplectic involutions

Connected group scheme smoothness with cohomological automorphisms

Reformulated Harder's Gauss-Bonnet Theorem adelically

Abstract

The reduced norm-one group G of a central simple algebra is an inner form of the special linear group, and an involution on the algebra induces an automorphism of G. We study the action of such automorphisms in the cohomology of arithmetic subgroups of G. The main result is a precise formula for Lefschetz numbers of automorphisms induced by involutions of symplectic type. Our approach is based on a careful study of the smoothness properties of group schemes associated with orders in central simple algebras. Along the way we also derive an adelic reformulation of Harder's Gauss-Bonnet Theorem.