717,519 research outputs found

    On the distribution of class groups of number fields

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    We propose a modification of the predictions of the Cohen--Lenstra heuristic for class groups of number fields in the case where roots of unity are present in the base field. As evidence for this modified formula we provide a large set of computational data which show close agreement.Comment: 14 pages. To appear in Experimental Mat

    Principalization algorithm via class group structure

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    For an algebraic number field K with 3-class group Cl3(K)Cl_3(K) of type (3,3), the structure of the 3-class groups Cl3(Ni)Cl_3(N_i) of the four unramified cyclic cubic extension fields NiN_i, 1i41\le i\le 4, of K is calculated with the aid of presentations for the metabelian Galois group G32(K)=Gal(F32(K)K)G_3^2(K)=Gal(F_3^2(K) | K) of the second Hilbert 3-class field F32(K)F_3^2(K) of K. In the case of a quadratic base field K=Q(D)K=\mathbb{Q}(\sqrt{D}) it is shown that the structure of the 3-class groups of the four S3S_3-fields N1,,N4N_1,\ldots,N_4 frequently determines the type of principalization of the 3-class group of K in N1,,N4N_1,\ldots,N_4. This provides an alternative to the classical principalization algorithm by Scholz and Taussky. The new algorithm, which is easily automatizable and executes very quickly, is implemented in PARI/GP and is applied to all 4596 quadratic fields K with 3-class group of type (3,3) and discriminant 106<D<107-10^6<D<10^7 to obtain extensive statistics of their principalization types and the distribution of their second 3-class groups G32(K)G_3^2(K) on various coclass trees of the coclass graphs G(3,r), 1r61\le r\le 6, in the sense of Eick, Leedham-Green, and Newman.Comment: 33 pages, 2 figures, presented at the Joint CSASC Conference, Danube University, Krems, Austria, September 201

    Distribution of the bad part of the class group

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    The Cohen-Lenstra-Martinet Heuristics gives a prediction of the distribution of ClK[p]\operatorname{Cl}_K[p^\infty] whne KK runs over Γ\Gamma-fields and pΓp\nmid|\Gamma|. In this paper, we prove several results on the distribution of ideal class groups for some pΓp||\Gamma|, and show that the behaviour is qualitatively different than what is predicted by the heuristics when pΓp\nmid|\Gamma|.We do this by using genus theory and the invariant part of the class group to investigate the algebraic structure of the class group. For general number fields, our result is conditional on a natural conjecture on counting fields. For abelian or D4D_4-fields, our result is unconditional.Comment: All comments are welcome

    Conjectures for distributions of class groups of extensions of number fields containing roots of unity

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    Cohen, Lenstra, and Martinet have given conjectures for the distribution of class groups of extensions of number fields, but Achter and Malle have given theoretical and numerical evidence that these conjectures are wrong regarding the Sylow pp-subgroups of the class group when the base number field contains ppth roots of unity. We give complete conjectures of the distribution of Sylow pp-subgroups of class groups of extensions of a number field when pp does not divide the degree of the Galois closure of the extension. These conjectures are based on qq\rightarrow\infty theorems on these distributions in the function field analog and use recent work of the authors on explicitly giving a distribution of modules from its moments. Our conjecture matches many, but not all, of the previous conjectures that were made in special cases taking into account roots of unity.Comment: 35 page
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