Gauss's Class Number Conjecture

In his monumental treatise Disquisitiones Arithmeticae, Gauß conjectured that the Class Number $h(-d)$ of an Imaginary quadratic field with Discriminant $-d$ tends to infinity with $d$. A proof was finally given by Heilbronn (1934), and Siegel (1936) showed that for any $\epsilon>0$, there exists a constant $c_\epsilon>0$ such that

$$h(-d)>c_\epsilon d^{1/2-\epsilon}$$

as $d\to\infty$. However, these results were not effective in actually determining the values for a given $m$ of a complete list of fundamental discriminants $-d$ such that $h(-d)=m$, 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 $s=1$, then for any $\epsilon>0$, there exists an effectively computable constant $c_\epsilon$ such that

$$h(-d)>c_\epsilon(\ln d)^{1-\epsilon}.$$

Gross and Zaiger (1983) showed that certain curves must satisfy the condition of Goldfeld, and Goldfeld's proof was simplified 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 $L$.'' 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.