Class numbers of quadratic function fields and continued fractions
نویسندگان
چکیده
منابع مشابه
Class numbers of ray class fields of imaginary quadratic fields
Let K be an imaginary quadratic field with class number one and let p ⊂ OK be a degree one prime ideal of norm p not dividing 6dK . In this paper we generalize an algorithm of Schoof to compute the class numbers of ray class fields Kp heuristically. We achieve this by using elliptic units analytically constructed by Stark and the Galois action on them given by Shimura’s reciprocity law. We have...
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We construct all families of quartic polynomials over Qwhose square root has a periodic continued fraction expansion, and detail those expansions. In particular we prove that, contrary to expectation, the cases of period length nine and eleven do not occur. We conclude by providing a list of examples of pseudo-elliptic integrals involving square roots of polynomials of degree four. The primary ...
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Let d ≥ 2 be a square-free integer and for all n ≥ 0, let l( √ d 2n+1 ) be the length of the continued fraction expansion of √ d 2n+1 . If Q( √ d) is a principal quadratic field, then under a condition on the fundamental unit of Z[ √ d] we prove that there exist constants C1 and C2 such that C1 √ d 2n+1 ≥ l( √ d 2n+1 ) ≥ C2 √ d 2n+1 for all large n. This is a generalization of a theorem of S. C...
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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 ...
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It is widely believed that the continued fraction expansion of every irrational algebraic number α either is eventually periodic (and we know that this is the case if and only if α is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine in [22] (see also [6,39,41] for surveys including a discussion on this subj...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1992
ISSN: 0022-314X
DOI: 10.1016/0022-314x(92)90027-m