Cuspidal cohomology classes for GL_n(Z)

TL;DR

This paper proves the existence of specific cuspidal automorphic representations for high-dimensional general linear groups and constructs new cuspidal cohomology classes, advancing understanding in automorphic forms and cohomology.

Contribution

It introduces a method to construct cuspidal automorphic representations for GL_n(Z) using symmetric power functoriality and Galois deformation theory, leading to new cohomology classes.

Findings

Existence of a cuspidal automorphic representation for GL_{79}/Q of level one and weight zero.

First known construction of cuspidal cohomology classes in H^*(GL_n(Z), C) for n > 1.

Application of symmetric power functoriality and Galois deformation theory in automorphic form construction.

Abstract

We prove the existence of a cuspidal automorphic representation $\pi$ for $GL_{79}/\mathbf{Q}$ of level one and weight zero. We construct $\pi$ using symmetric power functoriality and a change of weight theorem, using Galois deformation theory. As a corollary, we construct the first known cuspidal cohomology classes in $H^*(GL_{n}(\mathbf{Z}),\mathbf{C})$ for any $n > 1$.