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The Cohen-Lenstra heuristic: methodology and results.(English)Zbl 1213.11167

H. Cohen andH. W. Lenstra jun. conjectured that the sequence of class groups of quadratic number fields behaves like a random sequence with respect to a certain probability distribution on the space of all finite abelian groups [Number theory, Proc. Journ. arith., Noodwijkerhout/Neth. 1983, Lect. Notes Math. 1068, 33–62 (1984;Zbl 0558.12002)]. This probability distribution is based on the heuristic that probability for a group to occur should be inverse proportional to the number of its automorphisms. On the other hand, group theorists have studied a probability measure on the set of all partitions induced by the probability that a randomly chosen \(n\times n\)-matrix over \({\mathbb F}_p\) is contained in a conjugacy class associated with these partitions, for \(n\to\infty\). The two probability measures are connected to each other. The author gives an overview of the state of art.

MSC:

11M50 Relations with random matrices
11R29 Class numbers, class groups, discriminants
20P05 Probabilistic methods in group theory
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization

Citations:

Zbl 0558.12002

Cite

References:

[1]Achter, Jeffrey D., The distribution of class groups of function fields, J. Pure Appl. Algebra, 204, 2, 316-333 (2006) ·Zbl 1134.11042
[2]Andrews, George E., The Theory of Partitions, Encyclopedia Math. Appl., vol. 2 (1976), Addison-Wesley Publishing Co.: Addison-Wesley Publishing Co. Reading, MA, London, Amsterdam, MR 0557013 (58 #27738) ·Zbl 0371.10001
[3]Bennett, Curtis; Dempsey, Kathy J.; Sagan, Bruce E., Partition lattice \(q\)-analogs related to \(q\)-Stirling numbers, J. Algebraic Combin., 3, 3, 261-283 (1994), MR 1285496 (95h:05014) ·Zbl 0849.05004
[4]Cohen, H.; Lenstra, H. W., Heuristics on class groups of number fields, (Number Theory. Number Theory, Noordwijkerhout, 1983. Number Theory. Number Theory, Noordwijkerhout, 1983, Lecture Notes in Math., vol. 1068 (1984), Springer: Springer Berlin), 33-62, MR 756082 (85j:11144) ·Zbl 0558.12002
[5]Cohen, Henri, On the \(p^k\)-rank of finite abelian groups and Andrews’ generalizations of the Rogers-Ramanujan identities, Nederl. Akad. Wetensch. Indag. Math., 47, 4, 377-383 (1985), MR MR820930 (87g:20091) ·Zbl 0581.05008
[6]Cohen, Henri; Martinet, Jacques, Class groups of number fields: numerical heuristics, Math. Comp., 48, 177, 123-137 (1987), MR 866103 (88e:11112) ·Zbl 0627.12006
[7]Cohen, Henri; Martinet, Jacques, Étude heuristique des groupes de classes des corps de nombres, J. Reine Angew. Math., 404, 39-76 (1990), MR 1037430 (91k:11097) ·Zbl 0699.12016
[8]Evans, Steven N., Elementary divisors and determinants of random matrices over a local field, Stochastic Process. Appl., 102, 1, 89-102 (2002), MR MR1934156 (2004c:15041) ·Zbl 1075.15500
[9]Friedman, Eduardo; Washington, Lawrence C., On the distribution of divisor class groups of curves over a finite field, (Théorie des nombres. Théorie des nombres, Quebec, PQ, 1987 (1989), de Gruyter: de Gruyter Berlin), 227-239, MR 1024565 (91e:11138) ·Zbl 0693.12013
[10]Jason Fulman, Probability in the classical groups over finite fields: symmetric functions, stochastic algorithms and cycle indices, PhD thesis, Harvard University, 1997; Jason Fulman, Probability in the classical groups over finite fields: symmetric functions, stochastic algorithms and cycle indices, PhD thesis, Harvard University, 1997
[11]Fulman, Jason, A probabilistic approach toward conjugacy classes in the finite general linear and unitary groups, J. Algebra, 212, 2, 557-590 (1999), MR 1676854 (2000c:20072) ·Zbl 0980.20036
[12]Fulman, Jason, The Rogers-Ramanujan identities, the finite general linear groups, and the Hall-Littlewood polynomials, Proc. Amer. Math. Soc., 128, 1, 17-25 (2000), MR 1657747 (2000h:05229) ·Zbl 1005.11050
[13]Fulman, Jason, A probabilistic proof of the Rogers-Ramanujan identities, Bull. Lond. Math. Soc., 33, 4, 397-407 (2001), MR MR1832551 (2002b:11146) ·Zbl 1040.11074
[14]Jason Fulman, Email contact, 2008; Jason Fulman, Email contact, 2008
[15]Gauß, Carl Friedrich, Disquisitiones arithmeticae (1801), Springer: Springer Berlin, Leipzig ·Zbl 0136.32301
[16]Gerstenhaber, Murray, On the number of nilpotent matrices with coefficients in a finite field, Illinois J. Math., 5, 330-333 (1961), MR 0130875 (24 #A729) ·Zbl 0102.01602
[17]Hall, P., A partition formula connected with abelian groups, Comment. Math. Helv., 11, 1, 126-129 (1938) ·Zbl 0019.39705
[18]Kung, Joseph P. S., The cycle structure of a linear transformation over a finite field, Linear Algebra Appl., 36, 141-155 (1981), MR 604337 (82d:15012) ·Zbl 0477.05008
[19]Lengler, Johannes, A combinatorial interpretation of the probabilities of \(p\)-groups in the Cohen-Lenstra measure, J. Number Theory, 128, 7, 2070-2084 (2008), MR 2423750 ·Zbl 1180.05125
[20]Johannes Lengler, The Cohen-Lenstra heuristic for finite abelian groups, PhD thesis, Universität des Saarlandes, Saarbrücken, Germany, 2009; Johannes Lengler, The Cohen-Lenstra heuristic for finite abelian groups, PhD thesis, Universität des Saarlandes, Saarbrücken, Germany, 2009 ·Zbl 1294.11200
[21]Malle, Gunter, The totally real primitive number fields of discriminant at most \(10^9\), (Algorithmic Number Theory. Algorithmic Number Theory, Lecture Notes in Comput. Sci., vol. 4076 (2006), Springer: Springer Berlin), 114-123, MR 2282919 (2007j:11179) ·Zbl 1143.11371
[22]Gunter Malle, On the distribution of class groups of number fields, 2009; Gunter Malle, On the distribution of class groups of number fields, 2009 ·Zbl 1297.11139
[23]Bernd Mehnert, PhD thesis, Universität des Saarlandes, Germany, in press; Bernd Mehnert, PhD thesis, Universität des Saarlandes, Germany, in press
[24]Arunas Rudvalis, Ken-ichi Shinoda, An enumeration in finite classical groups, preprint, Department of Mathematics, U-Mass Amherst, 1988; Arunas Rudvalis, Ken-ichi Shinoda, An enumeration in finite classical groups, preprint, Department of Mathematics, U-Mass Amherst, 1988
[25]Stong, Richard, Some asymptotic results on finite vector spaces, Adv. in Appl. Math., 9, 2, 167-199 (1988), MR 937520 (89c:05007) ·Zbl 0681.05004
[26]Stong, Richard, The average order of a matrix, J. Combin. Theory Ser. A, 64, 2, 337-343 (1993), MR 1245166 (94j:11094) ·Zbl 0790.05019
[27]Washington, Lawrence C., Some remarks on Cohen-Lenstra heuristics, Math. Comp., 47, 176, 741-747 (1986) ·Zbl 0627.12002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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