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Colloquium

Congruent numbers, quadratic forms and algebraic K-theory

  • Speaker: QIN Hourong (Nanjing University)

  • Time: Oct 11, 2019, 16:30-17:30

  • Location: Conference Room 415, Hui Yuan 3#

Abstract

We show that if  a square-free and odd (respectively, even) positive integer is a congruent number, then $$\#\{(x,y,z) \in \mathbb{Z}^3|n=x^2+2y^2+32z^2\}=\#\{(x,y,z) \in \mathbb{Z}^3|n=2x^2+4y^2+9z^2-4yz\},$$\#\{(x,y,z) \in \mathbb{Z}^3|\frac{n}{2}=x^2+4y^2+32z^2\}=\#\{(x,y,z) \in \mathbb{Z}^3|\frac{n}{2}=4x^2+4y^2+9z^2-4yz\}.$$

If we assume that the weak Brich-Swinnerton-Dyer conjecture is true for the elliptic curves , then, conversely, these equalities imply that is a congruent number.

We shall also discuss some applications of the proposed method. In particular, for a prime , we show that  if   is a congruent number, then the -rank of equals one  and if with then is not a congruent number.