Abstract
We are interested in the question of counting cuspidal and selfdual automorphic representations of GL(n) over Q which are unramified at all primes 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 over Q unramified 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 lattices 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 ramified definite orthogonal groups to solve the n=26 case. A key ingredient is the new classification of (odd) unimodular lattices of rank 29, and of some nonunimodular (but even) lattices of lower rank.
