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H 1 Congruent number problem and BSD conjecture

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

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 Vidéo Langue : Anglais Date de publication : 09/06/16 Date de captation : 25/05/16 Collection : Research talks ; Number Theory Format : MP4 Durée : 00:49:00 Domaine : Number Theory Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2016-05-25_Zhang.mp4 Informations sur la rencontre Nom de la rencontre : Jean-Morlet Chair: Relative trace formula, periods, L-functions and harmonic analysis / Chaire Jean-Morlet : Formule des traces relatives, périodes, fonctions L et analyse harmoniqueOrganisateurs de la rencontre : Chaudouard, Pierre-Henri ; Heiermann, Volker ; Prasad, Dipendra ; Sakellaridis, YiannisDates : 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. - http://arxiv.org/abs/1603.08479v2

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

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