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Mathematics of Computation

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Class numbers of imaginary quadratic fields

Author: Mark Watkins
Journal: Math. Comp. 73 (2004), 907-938
MSC (2000): Primary 11R29; Secondary 11M06, 11Y35
Published electronically: October 2, 2003
MathSciNet review: 2031415
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Abstract: The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number $N$. The first complete results were for $N=1$ by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any $N$. Indeed, after Oesterlé handled $N=3$, 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 $N=5,6,7$, Wagner remarked that the $N=8$ case seemed impregnable. We complete the classification for all $N\le 100$, 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 $L$-function with an order 3 zero at the central critical point, to instead consider Dirichlet $L$-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

Received by editor(s): February 27, 2002
Published electronically: October 2, 2003
Article copyright: © Copyright 2003 American Mathematical Society