Cohen-Lenstra sums over local rings
Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 3, p. 817-838

We study series of the form M |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 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= p [C p k ] and u=0 converges. Moreover we consider refined sums, where M runs through all modules satisfying additional cohomological conditions.

On étudie des séries de la forme M |Aut R (M)| -1 |M| -u , où R est un anneau commutatif local et u est un entier non-negatif, la sommation s’étendant sur tous les R-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 R a des propriétés additionelles, on reliera les sommes ci-dessus à une limite de fonctions zêta des modules libres R n , ces fonctions zêta comptant les sous-R-modules d’indice fini dans R n . En particulier on montrera que cela est le cas pour l’anneau de groupe p [C p k ] d’un groupe cyclique d’ordre p k sur les entiers p-adiques. Par conséquant on pourra prouver une conjecture de [5], affirmant que la somme ci-dessus correspondante à R= p [C p k ] et u=0 converge. En outre on considère des sommes raffinées, où M parcourt tous les modules satisfaisant des conditions cohomologiques additionelles.

DOI : https://doi.org/10.5802/jtnb.471
Classification:  11S45,  16H05,  20C05
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     author = {Wittmann, Christian},
     title = {Cohen-Lenstra sums over local rings},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {3},
     year = {2004},
     pages = {817-838},
     doi = {10.5802/jtnb.471},
     mrnumber = {2144968},
     zbl = {02188542},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2004__16_3_817_0}
}
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://www.numdam.org/item/JTNB_2004__16_3_817_0/

[1] C.J. Bushnell, I. Reiner, Zeta functions of arithmetic orders and Solomon’s Conjectures. Math. Z. 173 (1980), 135–161. | MR 583382 | Zbl 0438.12004

[2] H. Cohen, H.W. Lenstra, Heuristics on class groups of number fields. Number Theory Noordwijkerhout 1983, LNM 1068, Springer, 1984. | MR 756082 | Zbl 0558.12002

[3] H. Cohen, J. Martinet, Étude heuristique des groupes de classes des corps de nombres. J. reine angew. Math. 404 (1990), 39–76. | MR 1037430 | Zbl 0699.12016

[4] S.D. Fisher, M.N. Alexander, Matrices over a finite field. Am. Math. Monthly 73 (1966), 639–641. | MR 1533848 | Zbl 0138.01202

[5] C. Greither, Galois-Cohen-Lenstra heuristics. Acta Math. et Inf. Univ. Ostraviensis 8 (2000), 33–43. | MR 1800220 | Zbl 1075.11070

[6] P. Hall, A partition formula connected with Abelian groups. Comment. Math. Helv. 11 (1938/39), 126–129. | MR 1509594 | Zbl 0019.39705

[7] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers. Oxford University Press, 1979. | MR 568909 | Zbl 0423.10001 | Zbl 0020.29201

[8] B. Huppert, Endliche Gruppen I. Springer, 1967. | MR 224703 | Zbl 0217.07201

[9] N. Jacobson, Basic Algebra II. Freeman, 1980. | MR 571884 | Zbl 0441.16001

[10] I. Reiner, Zeta functions of integral representations. Comm. Algebra 8 (1980), 911-925. | MR 573461 | Zbl 0444.12009

[11] G.-C. Rota, On the foundations of combinatorial theory I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie 2 (1964), 340–368. | MR 174487 | Zbl 0121.02406

[12] J.-P. Serre, Local Fields. Springer, 1995. | MR 554237 | Zbl 0423.12016

[13] L. Solomon, Zeta functions and integral representation theory. Adv. Math. 26 (1977), 306–326. | MR 460292 | Zbl 0374.20007

[14] C. Wittmann, Zeta functions of integral representations of cyclic p-groups. J. Algebra 274 (2004), 271–308. | MR 2040875 | Zbl 1052.20006

[15] C. Wittmann, p-class groups of certain extensions of degree p. Math. Comp. 74 (2005), 937–947. | MR 2114656 | Zbl 02140100