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1.
Let K be an imaginary abelian number field. We determine the unit index of K of certain type whose conductor is a product of two or three prime powers. As a consequence we construct some counterexamples to Satz 29 in Hasse's monograph [2]. 相似文献
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Let be an imaginary abelian number field. We know that , the relative class number of , goes to infinity as , the conductor of , approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CM-fields with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of . Second, we have proved in this paper that there are exactly 48 such fields.
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Louboutin Stéphane 《manuscripta mathematica》1996,91(1):343-352
Lately, I. Miyada proved that there are only finitely many imaginary abelian number fields with Galois groups of exponents
≤2 with one class in each genus. He also proved that under the assumption of the Riemann hypothesis there are exactly 301
such number fields. Here, we prove the following finiteness theorem: there are only finitely many imaginary abelian number
fields with one class in each genus. We note that our proof would make it possible to find an explict upper bound on the discriminants
of these number fields which are neither quadratic nor biquadratic bicyclic. However, we do not go into any explicit determination. 相似文献
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In this paper, we associate canonically to every imaginary quadratic field K= one or two isogenous classes of CM (complex multiplication) abelian varieties over K, depending on whether D is odd or even (D4). These abelian varieties are characterized as of smallest dimension and smallest conductor, and such that the abelian varieties themselves descend to . When D is odd or divisible by 8, they are the scalar restriction of canonical elliptic curves first studied by Gross and Rohrlich. We prove that these abelian varieties have the striking property that the vanishing order of their L-function at the center is dictated by the root number of the associated Hecke character. We also prove that the smallest dimension of a CM abelian variety over K is exactly the ideal class number of K and classify when a CM abelian variety over K has the smallest dimension.Mathematics Subject Classification (1991): 11G05, 11M20, 14H52Partially supported by a NSF grant DMS-0302043 相似文献
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Let be an abelian number field of degree . Most algorithms for computing the lattice of subfields of require the computation of all the conjugates of . This is usually achieved by factoring the minimal polynomial of over . In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of , which is based on -adic techniques. Given and a rational prime which does not divide the discriminant of , the algorithm computes the Frobenius automorphism of in time polynomial in the size of and in the size of . By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of .
8.
M. Hirabayashi 《Archiv der Mathematik》2002,79(3):175-181
We give a relative class number formula for an imaginary abelian number field by means of a determinant, which yields a generalization of Inkeris determinant. 相似文献
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Mathematische Annalen - 相似文献
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Tame and wild kernels of quadratic imaginary number fields 总被引:2,自引:0,他引:2
For all quadratic imaginary number fields of discriminant
we give the conjectural value of the order of Milnor's group (the tame kernel) where is the ring of integers of Assuming that the order is correct, we determine the structure of the group and of its subgroup (the wild kernel). It turns out that the odd part of the tame kernel is cyclic (with one exception, ).
we give the conjectural value of the order of Milnor's group (the tame kernel) where is the ring of integers of Assuming that the order is correct, we determine the structure of the group and of its subgroup (the wild kernel). It turns out that the odd part of the tame kernel is cyclic (with one exception, ).
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Ichiro Miyada 《manuscripta mathematica》1995,88(1):535-540
We show that there exist only finitely many imaginary abelian number fields of type (2,2,...,2) with one class in each genus.
Moreover, if the Generalized Riemann Hypothesis is true, we have exactly 301 such fields, whose degrees are less than or equal
to 23. Finally we give the table of those 301 fields. 相似文献
16.
Stéphane R. Louboutin 《Journal of Number Theory》2009,129(10):2289-2294
J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants OnoK are equal to their class numbers hK. Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if OnoK is large enough compared with hK, then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values dK of their discriminants are less than 2.3⋅109. We determine all these K's with dK?106. There are 76 such K's. We prove that there is at most one such K with dK?1.8⋅1011. 相似文献
17.
J. H. E. Cohn 《Proceedings of the American Mathematical Society》2002,130(5):1275-1277
Theorem. Let 2$"> denote an integer, the square-free part of and the class number of the field . Then except for the case , divides .
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Mikihito Hirabayashi 《manuscripta mathematica》2002,109(2):223-227
We give a relative class number formula for an imaginary abelian number field by means of some ``Dedekind sum'. This is
a generalization of Carlitz and Olson's formula presented in 1955.
Received: 18 December 2001 / Revised version: 15 May 2002
Mathematics Subject Classification (2000): 11R29, 11R20 相似文献
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Arnold Pizer 《Journal of Number Theory》1976,8(2):184-192
We employ a type number formula from the theory of quaternion algebras to gain information on the 2-part of the class numbers of imaginary quadratic number fields whose discriminants are divisible by three or fewer prime numbers. 相似文献