Abstract
We are interested in the question of counting cuspidal and autodual automorphic representations of GL(n) over Q which are unbranched in all prime and algebraic numbers of given distinct weights. Thanks to Arthur's work, it essentially amounts to the same thing to count the multiplicities of real discrete series in the spaces of automorphic forms of level 1of classical groups on Q unbranched everywhere, a problem however equally difficult. In recent years, this approach has made it possible to give exact formulas for this counting up to dimension n=24 (Chenevier-Renard, Taïbi, Chenevier-Taïbi). Potential applications to the cuspidal cohomology of GL_n(Z) and to Euclidean networks make us want to push these calculations a little further. In this talk, I will briefly revisit the above work, and then try to explain a recent work with Olivier Taïbi in which we manage to use brancheddefinite orthogonal groups to solve the n=26 case. A key ingredient is the new classification of unimodular (odd) networks of dimension 29, and of some nonunimodular (but even) networks of lower rank.
