Class numbers of imaginary quadratic fields

Author:
Mark Watkins

Journal:
Math. Comp. **73** (2004), 907-938

MSC (2000):
Primary 11R29; Secondary 11M06, 11Y35

DOI:
https://doi.org/10.1090/S0025-5718-03-01517-5

Published electronically:
October 2, 2003

MathSciNet review:
2031415

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Abstract | References | Similar Articles | Additional Information

Abstract: The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number . The first complete results were for by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any . Indeed, after Oesterlé handled , in 1985 Serre wrote, ``No doubt the same method will work for other small class numbers, up to 100, say.'' However, more than ten years later, after doing , Wagner remarked that the case seemed impregnable. We complete the classification for all , an improvement of four powers of 2 (arguably the most difficult case) over the previous best results. The main theoretical technique is a modification of the Goldfeld-Oesterlé work, which used an elliptic curve -function with an order 3 zero at the central critical point, to instead consider Dirichlet -functions with low-height zeros near the real line (though the former is still required in our proof). This is numerically much superior to the previous method, which relied on work of Montgomery-Weinberger. Our method is still quite computer-intensive, but we are able to keep the time needed for the computation down to about seven months. In all cases, we find that there is no abnormally large ``exceptional modulus'' of small class number, which agrees with the prediction of the Generalised Riemann Hypothesis.

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Additional Information

**Mark Watkins**

Affiliation:
Department of Mathematics, McAllister Building, The Pennsylvania State University, University Park, Pennsylvania 16802

Email:
watkins@math.psu.edu

DOI:
https://doi.org/10.1090/S0025-5718-03-01517-5

Received by editor(s):
February 27, 2002

Published electronically:
October 2, 2003

Article copyright:
© Copyright 2003
American Mathematical Society