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Some remarks on Cohen-Lenstra heuristics

Author: Lawrence C. Washington
Journal: Math. Comp. 47 (1986), 741-747
MSC: Primary 11R11; Secondary 11Y40
MathSciNet review: 856717
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Abstract: Cohen and Lenstra have given a heuristic model which predicts the fraction of imaginary quadratic fields with class number divisible by a given odd prime p and of those whose class groups have a given p-rank. We show that these numbers also arise by considering the proportion of matrices in $ {\text{GL}_n}({\mathbf{Z}}/p{\mathbf{Z}})$ with 1 as an eigenvalue and those whose 1-eigenspaces have a given dimension, then letting $ n \to \infty $. In the last section we discuss some relations with elliptic curves.

References [Enhancements On Off] (What's this?)

  • [1] H. Cohen & H. W. Lenstra, Jr., "Heuristics on class groups of number fields," Number Theory Noordwijkerhout. 1983 (H. Jager, ed.), Lecture Notes in Math., vol. 1068, Springer-Verlag, Berlin and New York, 1984, pp. 33-62. MR 756082 (85j:11144)
  • [2] G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, London, 1968.
  • [3] J.-P. Serre, "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques," Invent. Math., v. 15, 1972, pp. 259-331. MR 0387283 (52:8126)

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Article copyright: © Copyright 1986 American Mathematical Society

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