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H 1 Class number statistics for imaginary quadratic fields

Auteurs : Kurlberg, Pär (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : The number $F(h)$ of imaginary quadratic fields with class number $h$ is of classical interest: Gauss’ class number problem asks for a determination of those fields counted by $F(h)$. The unconditional computation of $F(h)$ for $h \le 100$ was completed by M. Watkins, and K. Soundararajan has more recently made conjectures about the order of magnitude of $F(h)$ as $h \to \infty$ and determined its average order.
    For odd $h$ we refine Soundararajan’s conjecture to a conjectural asymptotic formula and also consider the subtler problem of determining the number $F(G)$ of imaginary quadratic fields with class group isomorphic to a given finite abelian group $G$.
    Using Watkins’ tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance $(\mathbb{Z}/3\mathbb{Z})^3$ does not). This observation is explained in part by the Cohen-Lenstra heuristics, which have often been used to study the distribution of the p-part of an imaginary quadratic class group. We combine the Cohen-Lenstra heuristics with a refinement of Soundararajan’s conjecture to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of “missing” class groups, for families of $p$-groups as $p$ tends to infinity. For instance, it appears that no groups of the form $(\mathbb{Z}/p\mathbb{Z})^3$ and $p$ prime occurs as a class group of a quadratic imaginary field.
    Conditionally on the Generalized Riemann Hypothesis, we extend Watkins’ data, tabulating $F(h)$ for odd $h \le 10^6$ and $F(G)$ for $G$ a $p$-group of odd order with $|G| \le 10^6$. (To do this, we examine the class numbers of all negative prime fundamental discriminants $-q$, for $q \le 1.1881 \cdot 10^{15}.$) The numerical evidence matches quite well with our conjectures.
    This is joint work with S. Holmin, N. Jones, C. McLeman, and K. Petersen.

    Codes MSC :
    11R11 - Quadratic extensions
    11R29 - Class numbers, class groups, discriminants

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 13/04/16
      Date de captation : 31/03/16
      Collection : Research talks ; Number Theory
      Format : MP4
      Durée : 00:42:17
      Domaine : Number Theory
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2016-03-31_kurlberg.mp4

    Informations sur la rencontre

    Nom de la rencontre : Dynamics and graphs over finite fields: algebraic, number theoretic and algorithmic aspects / Dynamique et graphes sur les corps finis : aspects algebriques, arithmétiques et algorithmiques
    Organisateurs de la rencontre : Chang, Mei-Chu ; von zur Gathen, Joachim ; Ostafe, Alina ; Pappalardi, Francesco
    Dates : 29/03/16 - 02/04/16
    Année de la rencontre : 2016
    URL Congrès : http://conferences.cirm-math.fr/1391.html

    Citation Data

    DOI : 10.24350/CIRM.V.18952303
    Cite this video as: Kurlberg, Pär (2016). Class number statistics for imaginary quadratic fields. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18952303
    URI : http://dx.doi.org/10.24350/CIRM.V.18952303


    Voir aussi

    Bibliographie

    1. Holmin, S., Jones, N., Kurlberg, P., McLeman, C., Petersen, & K.L. (2015). Missing class groups and class number statistics for imaginary quadratic fields. - http://arxiv.org/abs/1510.04387

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