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Class numbers of imaginary quadratic fields

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 Oesterle 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 < 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-Oesterle 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.