“Gauss's Class Number Conjecture”的概念、定义、习题解题方法及翻译-数学辞典

单词

Gauss's Class Number Conjecture

释义

Gauss's Class Number Conjecture

In his monumental treatise Disquisitiones Arithmeticae, Gauß conjectured that the Class Number of an Imaginary quadratic field with Discriminant tends to infinity with . A proof was finally given by Heilbronn (1934), and Siegel (1936) showed that forany , there exists a constant such that

. However, these results were not effective in actually determining the values for a given

of a completelist 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 computableconstant such that

Gross and Zaiger (1983) showed that certain curves must satisfy the condition of Goldfeld, and Goldfeld's proof wassimplified by Oesterlé (1985).

See also Class Number, Gauss's Class Number Problem, Heegner Number

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 .'' C. R. 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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