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$ L$-functions and class numbers of imaginary quadratic fields and of quadratic extensions of an imaginary quadratic field

Author: Stéphane Louboutin
Journal: Math. Comp. 59 (1992), 213-230
MSC: Primary 11R29; Secondary 11R16
MathSciNet review: 1134735
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Abstract: Starting from the analytic class number formula involving its L-function, we first give an expression for the class number of an imaginary quadratic field which, in the case of large discriminants, provides us with a much more powerful numerical technique than that of counting the number of reduced definite positive binary quadratic forms, as has been used by Buell in order to compute his class number tables. Then, using class field theory, we will construct a periodic character $ \chi $, defined on the ring of integers of a field K that is a quadratic extension of a principal imaginary quadratic field k, such that the zeta function of K is the product of the zeta function of k and of the L-function $ L(s,\chi )$. We will then determine an integral representation of this L-function that enables us to calculate the class number of K numerically, as soon as its regulator is known. It will also provide us with an upper bound for these class numbers, showing that Hua's bound for the class numbers of imaginary and real quadratic fields is not the best that one could expect. We give statistical results concerning the class numbers of the first 50000 quadratic extensions of $ {\mathbf{Q}}(i)$ with prime relative discriminant (and with K/Q a non-Galois quartic extension). Our analytic calculation improves the algebraic calculation used by Lakein in the same way as the analytic calculation of the class numbers of real quadratic fields made by Williams and Broere improved the algebraic calculation consisting in counting the number of cycles of reduced ideals. Finally, we give upper bounds for class numbers of K that is a quadratic extension of an imaginary quadratic field k which is no longer assumed to be of class number one.

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  • [1] H. Amara, Groupe des classes et unité fondamentale des extensions quadratiques relatives d'un corps quadratique imaginaire principal, Pacific J. Math. 96 (1981), 1-12. MR 634758 (84a:12006)
  • [2] P. Barrucand, H. C. Williams, and L. Baniuk, A computational technique for determining the class number of a pure cubic field, Math. Comp. 30 (1976), 312-323. MR 0392913 (52:13726)
  • [3] P. Barrucand, J. Loxton, and H. C. Williams, Some explicit upper bounds on the class number and regulator of a cubic field with negative discriminant, Pacific J. Math. 128 (1987), 209-222. MR 888515 (88e:11107)
  • [4] Duncan A. Buell, Class groups of quadratic fields, Math. Comp. 30 (1976), 610-623. MR 0404205 (53:8008)
  • [5] H. Cohen and J. Martinet, Class groups of number fields: numerical heuristics, Math. Comp. 48 (1987), 123-137. MR 866103 (88e:11112)
  • [6] H. Cohn, A classical invitation to algebraic numbers and class fields, Universitext, Springer-Verlag, New York, 1978. MR 506156 (80c:12001)
  • [7] P. G. L. Dirichlet, Recherche sur les formes quadratiques à coefficients et à indéterminée complexes, Werke I, pp. 533-618.
  • [8] L. J. Goldstein, Analytic number theory, Prentice-Hall, Englewood Cliffs, NJ, 1971. MR 0498335 (58:16471)
  • [9] E. Hecke, Lectures on the theory of algebraic numbers, Graduate Texts in Math., vol. 77, Springer-Verlag, New York, 1981. MR 638719 (83m:12001)
  • [10] Hua Loo Keng, Introduction to number theory, Springer-Verlag, Berlin, 1982. MR 665428 (83f:10001)
  • [11] S. Iyanaga, The theory of numbers, North-Holland Math. Library, North-Holland, Amsterdam, 1975.
  • [12] R. B. Lakein, Computation of the ideal class group of certain complex quartic fields, Math. Comp. 28 (1974), 839-846. MR 0374090 (51:10290)
  • [13] S. Louboutin, Minorations (sous l'hypothèse de Riemann généralisée) des nombres de classes des corps quadratiques imaginaires. Application, C. R. Acad. Sci. Paris Sér. I 310 (1990), 795-800. MR 1058499 (91e:11126)
  • [14] -, Nombres de classes d'idéaux des extensions quadratiques du corps de Gauss, preprint.
  • [15] -, Norme relative de l'unité fondamentale et 2-rang du groupe des classes d'idéaux de certains corps biquadratiques, Acta Arith. 58 (1991), 273-288. MR 1121087 (93a:11090)
  • [16] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, PWN, Warszawa, 1974. MR 0347767 (50:268)
  • [17] A. Ogg, Modular forms and Dirichlet series, Benjamin, New York, 1969. MR 0256993 (41:1648)
  • [18] J. J. Payan, Majoration du nombre de classes d'un corps cubique cyclique de conducteur premier, J. Math. Soc. Japan 33 (1981), 701-706. MR 630633 (83a:12006)
  • [19] J. P. Serre, Local class field theory, Ch. VI, 4.4, Global conductors, Algebraic Number Theory (J. W. S. Cassels and A. Fröhlich, eds.), Thompson, Washington D.C., 1967. MR 0220701 (36:3753)
  • [20] H. M. Stark, On complex quadratic fields with unit class number two, Math. Comp. 29 (1975), 289-302. MR 0369313 (51:5548)
  • [21] A. J. Stephens and H. C. Williams, Computation of real quadratic fields with class number one, Math. Comp. 51 (1988), 809-824. MR 958644 (90b:11106)
  • [22] H. C. Williams and J. Broere, A computational technique for evaluating $ L(1,\chi )$ and the class number of a real quadratic field, Math. Comp. 30 (1976), 887-893. MR 0414522 (54:2623)

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Article copyright: © Copyright 1992 American Mathematical Society

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