Some remarks on Cohen-Lenstra heuristics
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- by Lawrence C. Washington PDF
- Math. Comp. 47 (1986), 741-747 Request permission
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
- H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33–62. MR 756082, DOI 10.1007/BFb0099440 G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, London, 1968.
- Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI 10.1007/BF01405086
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Math. Comp. 47 (1986), 741-747
- MSC: Primary 11R11; Secondary 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-1986-0856717-9
- MathSciNet review: 856717