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 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 .'' *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.

© 1996-9

1999-05-25