Rational right triangles and rank distributions in families of elliptic curves

 A positive integer d is called a congruent number if there exists a right triangle with rational side lengths whose area is d.  After giving some background on the thousand-year history of this problem, we will prove that asymptotically 100% of the positive integers equal to 1, 2, or 3 mod 8 are not congruent numbers, and that 100% of the positive integers equal to 5, 6, or 7 mod 8 are congruent numbers (the latter following from recent work of D. Kriz).

To prove these results, we will study the variation in families of elliptic curves for the $2^k$-Selmer rank of an elliptic curve E over Q, these Selmer ranks being effectively computable upper bounds for the rank of the group of rational points of E.  As a consequence of this work, we will  show that asymptotically 100% of the quadratic twists of ‘most’ elliptic curves have Mordell-Weil rank less than 2.