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Exponents of Class Groups of Quadratic Function Fields over Finite Fields

  • David A. Cardon (a1) and M. Ram Murty (a2)
Abstract

We find a lower bound on the number of imaginary quadratic extensions of the function field whose class groups have an element of a fixed order.

More precisely, let q ≥ 5 be a power of an odd prime and let g be a fixed positive integer ≥ 3. There are polynomials D with deg(D) ≤ such that the class groups of the quadratic extensions have an element of order g.

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References
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[1] Artin, E., Quadratische Körper im Gebiet der höheren Kongruenzen I, II. Math. Z. 19 (1924), 153246.
[2] Ankeny, N. and Chowla, S., On the divisibility of the class numbers of quadratic fields. Pacific J. Math. 5 (1955), 321324.
[3] Cohen, H. and Lenstra, H. W. Jr., Heuristics on class groups of number fields. Number Theory (Noordwijkerhout, 1983) Proceedings, Springer Lecture Notes in Math. 1068, 1984.
[4] Davenport, H. and Heilbronn, H., On the density of discriminants of cubic fields, II. Proc. Royal Soc. London Ser. A 322 (1971), 405420.
[5] Gupta, R. and Ram Murty, M., Class groups of quadratic functions fields. In preparation.
[6] Friedman, Eduardo and Washington, Lawrence C., On the distribution of divisor class groups of curves over finite fields. In: Théorie des nombres (Quebec, PQ 1987), de Gruyter, Berlin, 1989, 227–239.
[7] Friesen, Christian, Class number divisibility in real quadratic function fields. Canad. Math. Bull. (3) 35 (1992), 361370.
[8] Hartung, P., Proof of the existence of infinitely many imaginary quadratic fields whose class number is not divisible by 3. J. Number Theory 6 (1974), 276278.
[9] Honda, T., A few remarks on class numbers of imaginary quadratic fields. Osaka J. Math. 12 (1975), 1921.
[10] Ram Murty, M., The ABC conjecture and exponents of class groups of quadratic fields. Contemp. Math. 210 (1998), 8595.
[11] Ram Murty, M., Exponents of class groups of quadratic fields. Topics in Number Theory (University Park, PA, 1997), Math. Appl. 467, Kluwer Acad. Publ., Dordrecht, 1999, 229239.
[12] Nagell, T., Über die Klassenzahl imaginär quadratischer Zahlkörper. Abh. Math. Sem. Univ. Hamburg, 1 (1922), 140150.
[13] Weinberger, P., Real Quadratic Fields with Class Numbers Divisible by n. J. Number Theory, 5 (1973), 237241.
[14] Yamamoto, Y., On unramified Galois extensions of quadratic number fields. Osaka J. Math. 7 (1970), 5776.
[15] Yu, Jiu-Kang, Toward the Cohen-Lenstra conjecture in the function field case. Preprint.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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