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Date: 13-8-2020
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Date: 18-4-2020
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In his monumental treatise Disquisitiones Arithmeticae, Gauss conjectured that the class number of an imaginary quadratic field with binary quadratic form discriminant
tends to infinity with
. A proof was finally given by Heilbronn (1934), and Siegel (1936) showed that for any
, there exists a constant
such that
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as . However, these results were not effective in actually determining the values for a given
of a complete list of fundamental discriminants
such that
, a problem known as Gauss's class number problem.
Goldfeld (1976) showed that if there exists a "Weil curve" whose associated Dirichlet L-series has a zero of at least third order at , then for any
, there exists an effectively computable constant
such that
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Gross and Zaiger (1983) showed that certain curves must satisfy the condition of Goldfeld, and Goldfeld's proof was simplified by Oesterlé (1985).
REFERENCES:
Arno, S.; Robinson, M. L.; and Wheeler, F. S. "Imaginary Quadratic Fields with Small Odd Class Number." http://www.math.uiuc.edu/Algebraic-Number-Theory/0009/.
Böcherer, S. "Das Gauß'sche Klassenzahlproblem." Mitt. Math. Ges. Hamburg 11, 565-589, 1988.
Gauss, C. F. Disquisitiones Arithmeticae. New Haven, CT: Yale University Press, 1966.
Goldfeld, D. M. "The Class Number of Quadratic Fields and the Conjectures of Birch and Swinnerton-Dyer." Ann. Scuola Norm. Sup. Pisa 3, 623-663, 1976.
Gross, B. and Zaiger, D. "Points de Heegner et derivées de fonctions ." Comptes Rendus Acad. Sci. Paris 297, 85-87, 1983.
Heilbronn, H. "On the Class Number in Imaginary Quadratic Fields." Quart. J. Math. Oxford Ser. 25, 150-160, 1934.
Oesterlé, J. "Nombres de classes des corps quadratiques imaginaires." Astérique 121-122, 309-323, 1985.
Siegel, C. L. "Über die Klassenzahl quadratischer Zahlkörper." Acta. Arith. 1, 83-86, 1936.
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