It is an open question to determine the first integer \(n>1\) such that the cuspidal cohomology of \(GL_n(\mathbb{Z})\) with rational coefficients is nonzero (we know \(27\leq n\leq 79\)). It is equivalent to ask for the existence of an algebraic cuspidal automorphic representation of \(GL_n\) over \(\mathbb{Q}\) which is unramified at all finite primes, and whose "weights" are the consecutive integers \(0,1,...,n-1\). What if we rather allow two holes in this string of consecutive weights? (necessarily some \(i\) and \(n+1-i\)) I will explain what is the smallest even integer n such that such a representation exists. A key role will be played by the classification of unimodular integral lattices of rank \(29\). Joint work with Olivier Taïbi.
Unimodular lattices and the cuspidal cohomology of \(GL_n(\mathbb{Z})\)
14.01.2025 13:15 - 14:45
Organiser:
H. Grobner, A. Minguez-Espallargas, A. Mellit
Location: