Cuspidal cohomology classes for 𝐺𝐿_{𝑛}(𝐙)

Abstract

We prove the existence of a cuspidal automorphic representation π \pi for GL 79 ⁡ / Q \operatorname {GL}_{79}/\mathbf {Q} of level one and weight zero. We construct π \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 ⁡ ( Z ) , C ) H^*(\operatorname {GL}_{n}(\mathbf {Z}),\mathbf {C}) for any n > 1 n > 1 .