Joint PU/IAS Number Theory

For an elliptic curve E defined over Q, the Mordell-Weil group E(Q) is a finitely generated abelian group. We prove that there are infinitely many elliptic curves E over Q for which E(Q) has rank 2. Our elliptic curves will be given by explicit models and their ranks will be found using a 2-descent. The infinitude of such elliptic curves will make use of a theorem of Tao and Ziegler. Time permitting we also describe some recent work on rank stability.