Nous définissons le -groupe des classes logarithmiques signées d’un corps de nombres par analogie avec le groupe des classes d’idéaux au sens restreint et nous établissons les résultats de base de l’arithmétique des classes logarithmiques signées.
We study series of the form , where is a commutative local ring, is a non-negative integer, and the summation extends over all finite -modules , up to isomorphism. This problem is motivated by Cohen-Lenstra heuristics on class groups of number fields, where sums of this kind occur. If has additional properties, we will relate the above sum to a limit of zeta functions of the free modules , where these zeta functions count -submodules of finite index in . In particular we will show that...
The aim of these notes is to generalize Laumon’s construction [20] of automorphic sheaves corresponding to local systems on a smooth, projective curve to the case of local systems with indecomposable unipotent ramification at a finite set of points. To this end we need an extension of the notion of parabolic structure on vector bundles to coherent sheaves. Once we have defined this, a lot of arguments from the article “ On the geometric Langlands conjecture” by Frenkel, Gaitsgory and Vilonen [11]...
Let be a prime and let be a -group of matrices in , for some integer . In this paper we show that, when , a certain subgroup of the cohomology group is trivial. We also show that this statement can be false when . Together with a result of Dvornicich and Zannier (see [2]), we obtain that any algebraic torus of dimension enjoys a local-global principle on divisibility by .