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Multi angle Congruent number problem and BSD conjecture

Auteurs : Zhang, Shou-Wu (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : A thousand years old problem is to determine when a square free integer $n$ is a congruent number ,i,e, the areas of right angled triangles with sides of rational lengths. This problem has a some beautiful connection with the BSD conjecture for elliptic curves $E_n : ny^2 = x^3 - x$. In fact by BSD, all $n= 5, 6, 7$ mod $8$ should be congruent numbers, and most of $n=1, 2, 3$ mod $8$ should not be congruent numbers. Recently, Alex Smith has proved that at least 41.9% of $n=1,2,3$ satisfy (refined) BSD in rank $0$, and at least 55.9% of $n=5,6,7$ mod $8$ satisfy (weak) BSD in rank $1$. This implies in particular that at last 41.9% of $n=1,2,3$ mod $8$ are not congruent numbers, and 55.9% of $n=5, 6, 7$ mod $8$ are congruent numbers. I will explain the ingredients used in Smith's proof: including the classical work of Heath-Brown and Monsky on the distribution F_2 rank of Selmer group of E_n, the complex formula for central value and derivative of L-fucntions of Waldspurger and Gross-Zagier and their extension by Yuan-Zhang-Zhang, and their mod 2 version by Tian-Yuan-Zhang.

    Codes MSC :
    11D25 - Cubic and quartic equations
    11G40 - $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
    11R29 - Class numbers, class groups, discriminants

    Informations sur la rencontre

    Nom du congrès : Jean-Morlet Chair: Relative trace formula, periods, L-functions and harmonic analysis / Chaire Jean-Morlet : Formule des traces relatives, périodes, fonctions L et analyse harmonique
    Organisteurs Congrès : Chaudouard, Pierre-Henri ; Heiermann, Volker ; Prasad, Dipendra ; Sakellaridis, Yiannis
    Dates : 23/05/2016 - 27/05/16
    Année de la rencontre : 2016
    URL Congrès : http://prasad-heiermann.weebly.com/main-...

    Citation Data

    DOI : 10.24350/CIRM.V.18981703
    Cite this video as: Zhang, Shou-Wu (2016). Congruent number problem and BSD conjecture.CIRM .Audiovisual resource. doi:10.24350/CIRM.V.18981703
    URI : http://dx.doi.org/10.24350/CIRM.V.18981703


    Voir aussi

    Bibliographie

    1. Smith, A. (2016). The congruent numbers have positive natural density. <arXiv:1603.08479> - http://arxiv.org/abs/1603.08479v2

    2. Tian, Y., Yuan, X., & Zhang, S.-W. (2014). Genus periods, genus points and congruent number problem. <arXiv:1411.4728> - https://arxiv.org/abs/1411.4728

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