Cohomology of braids, principal congruence subgroups and geometric representations

TL;DR

This paper computes the integral cohomology of principal congruence subgroups in SL(2,Z) and their braid group analogues with local coefficients, linking results to modular forms and homotopy theory.

Contribution

It extends previous work by providing explicit cohomology calculations with local coefficients, connecting algebraic, geometric, and topological aspects.

Findings

Cohomology related to modular forms in characteristic zero

Identification of torsion via p-divided power algebra

Comparison with classical Shimura computations

Abstract

The main purpose of this article is to give the integral cohomology of classical principal congruence subgroups in SL(2,Z) as well as their analogues in the third braid group with local coefficients in symmetric powers of the natural symplectic representation. The resulting answers (1) correspond to certain modular forms in characteristic zero, and (2) the cohomology of certain spaces in homotopy theory in characteristic p. The torsion is given in terms of the structure of a "p-divided power algebra". The work is an extension of the work in arXiv:1204.5390v1 as well as extensions of a classical computation of Shimura to integral coefficients. The results here contrast the local coefficients such as that in [Looijenga, J. Alg. Geom., 5, '96] and [Tillmann, Q. J. Math., 61, '10].