Computational Algebra Seminar
For each positive d, the elliptic curve d*y2 = x3 – x has positive rank if and only if d is the area of a right-angled triangle with rational sides, the latter being a question of antiquity. We do not address this question of positive rank per se, but rather present the results of some experiments trying to find d such that the above elliptic curve has large rank. These can be seen as d for which there exist "many" rational-sided triangles of area d.
Let X be a smooth double cover of a geometrically ruled surface over a separably closed field of characteristic different from 2. I will discuss recent work in which we give a finite presentation of the two-torsion in the Brauer group of X with generators given by central simple algebras over the function field of X and relations coming from the Neron-Severi group of X. I will also discuss some of the motivation for this coming from arithmetic applications such as computing Brauer-Manin obstructions to the existence of rational points. This is joint work with Bianca Viray.