Gauss's Class Number Conjecture

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

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

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.