Mathematics Colloquium

When does a cubic equation E: y^2=x^3+Ax+B (A,B integers) have finitely or infinitely many rational solutions? One answer is given in terms of an analytic criterion through the celebrated Birch-Swinnerton-Dyer conjecture (BSD for short). But there are other, algebraic criteria that should also be sufficient. This talk will introduce all these objects and describe some recent progress toward proving the sufficiency of these criteria and the validity of BSD