Shou-Wu Zhang: Congruent number problem and BSD conjecture

Abstract : 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 En:ny2=x3−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.

Recording during the thematic meeting: "Relative trace formula, periods, L-functions and harmonic analysis" the May 25, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Видео Shou-Wu Zhang: Congruent number problem and BSD conjecture канала Centre International de Rencontres Mathématiques