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(1966)
Hypoelliptic curves and the least prime quadratic residue
- Publisher
- de Gruyter
- Copyright
- © Walter de Gruyter
- ISSN
- 0933-7741
- eISSN
- 1435-5337
- DOI
- 10.1515/FORUM.2008.014
- Publisher site
- See Article on Publisher Site
Abstract
We show that there are finitely many imaginary quadratic number fields for which the class group has exponent 5. Indeed there are finitely many with exponent at most 6. The proof is based on a method of Pierce Pierce L. B.: A bound for the 3-part of class numbers of quadratic fields by means of the square sieve. Forum Math. 18 (2006), 677–698. The problem is reduced to one of counting integral points on a certain affine surface. This is tackled using the author's “square-sieve”, in conjunction with estimates for exponential sums. The latter are derived using the q -analogue of van der Corput's method. 2000 Mathematics Subject Classification: 11R29; 11D45, 11L07, 11N36.
Journal
Forum Mathematicum – de Gruyter
Published: Mar 1, 2008