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).

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and Swinnerton-Dyer." Ann. Scuola Norm. Sup. Pisa3, 623-663,
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Fields." Quart. J. Math. Oxford Ser.25, 150-160, 1934.Oesterlé,
J. "Nombres de classes des corps quadratiques imaginaires." Astérique121-122,
309-323, 1985.Siegel, C. L. "Über die Klassenzahl quadratischer
Zahlkörper." Acta. Arith.1, 83-86, 1936.