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 this is the case for the group ring of a cyclic group of order over the -adic integers. Thereby we are able to prove a conjecture from [5], stating that the above sum corresponding to and converges. Moreover we consider refined sums, where runs through all modules satisfying additional cohomological conditions.
On étudie des séries de la forme , où est un anneau commutatif local et est un entier non-negatif, la sommation s’étendant sur tous les -modules finis, à isomorphisme prés. Ce problème est motivé par les heuristiques de Cohen et Lenstra sur les groupes des classes des corps de nombres, où de telles sommes apparaissent. Si a des propriétés additionelles, on reliera les sommes ci-dessus à une limite de fonctions zêta des modules libres , ces fonctions zêta comptant les sous--modules d’indice fini dans . En particulier on montrera que cela est le cas pour l’anneau de groupe d’un groupe cyclique d’ordre sur les entiers -adiques. Par conséquant on pourra prouver une conjecture de [5], affirmant que la somme ci-dessus correspondante à et converge. En outre on considère des sommes raffinées, où parcourt tous les modules satisfaisant des conditions cohomologiques additionelles.
@article{JTNB_2004__16_3_817_0, author = {Wittmann, Christian}, title = {Cohen-Lenstra sums over local rings}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {817--838}, publisher = {Universit\'e Bordeaux 1}, volume = {16}, number = {3}, year = {2004}, doi = {10.5802/jtnb.471}, zbl = {02188542}, mrnumber = {2144968}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.471/} }
TY - JOUR AU - Wittmann, Christian TI - Cohen-Lenstra sums over local rings JO - Journal de théorie des nombres de Bordeaux PY - 2004 SP - 817 EP - 838 VL - 16 IS - 3 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.471/ DO - 10.5802/jtnb.471 LA - en ID - JTNB_2004__16_3_817_0 ER -
Wittmann, Christian. Cohen-Lenstra sums over local rings. Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 3, pp. 817-838. doi : 10.5802/jtnb.471. http://archive.numdam.org/articles/10.5802/jtnb.471/
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