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Class group frequencies
of real quadratic function fields:
The degree 4 case


Author: Christian Friesen
Journal: Math. Comp. 69 (2000), 1213-1228
MSC (1991): Primary 11R29, 11R58, 11R11
DOI: https://doi.org/10.1090/S0025-5718-99-01154-0
Published electronically: May 24, 1999
MathSciNet review: 1659859
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Abstract | References | Similar Articles | Additional Information

Abstract: The distribution of ideal class groups of $\mathbb{F}_{q}(T,\sqrt {M(T)})$ is examined for degree-four monic polynomials $M \in \mathbb{F}_{q}[T]$ when $\mathbb{F}_{q}$ is a finite field of characteristic greater than 3 with $q \in [20000,100000]$ or $q \in [1020000,1100000]$ and $M$ is irreducible or has an irreducible cubic factor. Particular attention is paid to the distribution of the $p$-Sylow part of the class group, and these results agree with those predicted using the Cohen-Lenstra heuristics to within about 1 part in 10000. An alternative set of conjectures specific to the cases under investigation is in even sharper agreement.


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  • 1. William W. Adams and Michael J. Razar, Multiples of points on elliptic curves and continued fractions, Proc. London Math. Soc. 41 (1980), 481-498. MR 82c:14031
  • 2. Emil Artin, Quadratische Körper im Gebiet der höheren Kongruenzen I, II, Math. Zeitschrift 19 (1924), 153-246.
  • 3. Duncan A. Buell, Class groups of quadratic fields II, Math. Comp. 48 (1987), 85-93. MR 87m:11109
  • 4. Duncan A. Buell, The expectation of success using a Monte Carlo factoring method - some statistics on quadratic class numbers, Math. Comp. 43 (1984), 313-327. MR 85k:11068
  • 5. H. Cohen and H. W. Lenstra, Jr., Heuristics on class groups of number fields, Number Theory Noordwijkerhout (H. Jager, ed.), Lecture Notes in Math. vol. 1068, Springer-Verlag, Berlin and New York, 1984, pp. 33-62. MR 85j:11144
  • 6. H. Cohen and J. Martinet, Class groups of finite fields: Numerical heuristics, Math. Comp 48 (1987), 123-137. MR 88e:11112
  • 7. M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272. MR 3:104f
  • 8. Eduardo Friedman and Lawrence C. Washington, On the distribution of divisor class groups of curves over a finite field, Théorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 227-239. MR 91c:11138
  • 9. Christian Friesen, A special case of Cohen-Lenstra heuristics in function fields, Fifth Conference of the Canadian Number Theory Association (Kenneth S. Williams and Rajiv Gupta, ed.), CRM Proceedings and Lecture Notes, vol. 19, American Mathematical Society, Providence, RI, 1999, pp. 99-105.
  • 10. KeQin Feng and Shu Ling Sun, On class number of quadratic function fields, Algebraic structures and number theory (Hong Kong 1988), World Sci. Publishing, Teaneck, NJ, 1990, pp. 88-113. MR 91m:11098
  • 11. S. Kuroda, Table of class numbers, $h(p) > 1$, for quadratic fields $Q(\sqrt {p})$, $p \equiv 1\ (\operatorname{mod} 4)\le 2776817$, Table, Univ. of Maryland, 1965, deposited in the UMT file; reviewed in Math. Comp. 29 (1975), 335-336.
  • 12. Michiyo Saito and Hideo Wada, Tables of ideal class groups of real quadratic fields, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), 347-349. MR 90a:11122
  • 13. R. Scheidler, A. Stein, H. C. Williams, Key-Exchange in Real Quadratic Congruence Function Fields, Des. Codes Cryptogr. 7 (1996), 153-174. MR 97d:94009
  • 14. René Schoof, Nonsingular Plane Cubic Curves over Finite Fields, Journal of Combinatorial Theory, Series A 46 (1987), 183-211. MR 88k:14013
  • 15. Andreas Stein, Equivalences between elliptic curves and real quadratic congruence function fields, J. Théor. Nombres Bordeaux 9 (1997), no. 1, 75-95. MR 98d:11144
  • 16. M. Tennenhouse and H. C. Williams, A note on class-number one in certain real quadratic and pure cubic fields, Math. Comp. 46 (1986), 333-336. MR 87b:11127
  • 17. E. Waterhouse, Abelian varieties over finite fields, Ann. Sci. Ecole Norm. Sup. (4) 2 (1969), 521-560. MR 42:279
  • 18. Jiu-Kang Yu, Toward a proof of the Cohen-Lenstra conjecture in the function field case, preprint, 1996.

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Additional Information

Christian Friesen
Affiliation: Ohio State University at Marion,1465 Mt. Vernon Ave, Marion, Ohio 43302
Email: friesen.4@osu.edu

DOI: https://doi.org/10.1090/S0025-5718-99-01154-0
Keywords: Class groups, Cohen-Lenstra conjecture, function fields, class numbers
Received by editor(s): September 8, 1998
Published electronically: May 24, 1999
Article copyright: © Copyright 2000 American Mathematical Society

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