Cohen-Lenstra sums over local rings

Wittmann, Christian. "Cohen-Lenstra sums over local rings." Journal de Théorie des Nombres de Bordeaux 16.3 (2004): 817-838. <http://eudml.org/doc/249250>.

@article{Wittmann2004,
abstract = {We study series of the form $\displaystyle \sum _M |\operatorname\{Aut\}_R(M)|^\{-1\} |M|^\{-u\}$, where $R$ is a commutative local ring, $u$ is a non-negative integer, and the summation extends over all finite $R$-modules $M$, up to isomorphism. This problem is motivated by Cohen-Lenstra heuristics on class groups of number fields, where sums of this kind occur. If $R$ has additional properties, we will relate the above sum to a limit of zeta functions of the free modules $R^n$, where these zeta functions count $R$-submodules of finite index in $R^n$. In particular we will show that this is the case for the group ring $\mathbb\{Z\}_p[C_\{p^k\}]$ of a cyclic group of order $p^k$ over the $p$-adic integers. Thereby we are able to prove a conjecture from [5], stating that the above sum corresponding to $R=\mathbb\{Z\}_p[C_\{p^k\}]$ and $u=0$ converges. Moreover we consider refined sums, where $M$ runs through all modules satisfying additional cohomological conditions.},
affiliation = {Universität der Bundeswehr München Fakultät für Informatik Institut für Theoretische Informatik und Mathematik 85577 Neubiberg, Germany},
author = {Wittmann, Christian},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {3},
pages = {817-838},
publisher = {Université Bordeaux 1},
title = {Cohen-Lenstra sums over local rings},
url = {http://eudml.org/doc/249250},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Wittmann, Christian
TI - Cohen-Lenstra sums over local rings
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 3
SP - 817
EP - 838
AB - We study series of the form $\displaystyle \sum _M |\operatorname{Aut}_R(M)|^{-1} |M|^{-u}$, where $R$ is a commutative local ring, $u$ is a non-negative integer, and the summation extends over all finite $R$-modules $M$, up to isomorphism. This problem is motivated by Cohen-Lenstra heuristics on class groups of number fields, where sums of this kind occur. If $R$ has additional properties, we will relate the above sum to a limit of zeta functions of the free modules $R^n$, where these zeta functions count $R$-submodules of finite index in $R^n$. In particular we will show that this is the case for the group ring $\mathbb{Z}_p[C_{p^k}]$ of a cyclic group of order $p^k$ over the $p$-adic integers. Thereby we are able to prove a conjecture from [5], stating that the above sum corresponding to $R=\mathbb{Z}_p[C_{p^k}]$ and $u=0$ converges. Moreover we consider refined sums, where $M$ runs through all modules satisfying additional cohomological conditions.
LA - eng
UR - http://eudml.org/doc/249250
ER -