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# Documents  Zhang, Shou-Wu | enregistrements trouvés : 3

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## Heegner points and Rankin L-series :papers from the workshop on special values of ... held at MSRI#Dec. Darmon, Henri ; Zhang, Shou-Wu | Cambridge University Press 2004

Congrès

- 367 p.
ISBN 978-0-521-83659-3

Mathematical sciences research institute publications , 0049

Localisation : Colloque 1er étage (BERK)

courbe elliptique # fonction L # série L # formule de Gross-Zagier # multiplication complexe # forme automorphe # méthode de Rankin-Selberg # intersection arithmétique # théorie d'Iwasawa # point de Heegner

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## Congruent number problem and BSD conjecture Zhang, Shou-Wu | CIRM H

Multi angle

Research talks;Number Theory

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. 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 ...

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## The Gross-Zagier formula on Shimura curves Yuan, Xinyi ; Zhang, Shou-Wu ; Zhang, Wei | Princeton University Press 2013

Ouvrage

- viii; 256 p.
ISBN 978-0-691-15591-3

Annals of mathematics studies , 0184

Localisation : Ouvrage RdC (YUAN)

variétés des Shimura # géométrie algébrique arithmétique # forme automorphe # quaternions

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