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Imaginary quadratic fields with small odd class number. (English) Zbl 0904.11031

In this paper the authors determine all imaginary quadratic number fields of odd class number \(h\) in the range \(5\leq h\leq 23\). The results for \(h=1,2,3,4\) were already known.
The authors employ techniques from Goldfeld, Oesterlé, Stark, Montgomery and Weinberger, and the first author. With those results it is straightforward to give upper bounds for the discriminant \(-d\) (\(d\equiv 3 \bmod 4\) a prime) of potential imaginary quadratic fields. The crucial part of the paper is the reduction of these huge bounds (\(10^{550}\) in the beginning) to a size of about \(10^{15}\). For this, the authors substantially refine an approach by S. Arno. They use a separation of the minima (minimal coefficients) of reduced quadratic forms of discriminant \(-d\) combined with earlier methods from Stark and Montgomery-Weinberger. The last step then consists of an enumeration procedure for determining all fields of given class number \(h\) below the reduced bound by sieving techniques.
Reviewer: M.Pohst (Berlin)

MSC:

11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11Y40 Algebraic number theory computations
11-04 Software, source code, etc. for problems pertaining to number theory
11H50 Minima of forms
11N36 Applications of sieve methods
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