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).
." 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.