Joint IAS/PU Number Theory Seminar

We prove that at least 2/21 of all integers can be written as a sum of two rational cubes, and at least 1/6 of all integers cannot. More generally, in any cubic twist family of elliptic curves, at least one 1/6 of curves have rank 0 and at least 1/6 of curves with good reduction at 2 have rank 1. I'll give an overview of the proof, which combines various methods (orbit-parameterizations over the integers, geometry-of-numbers, circle method, Iwasawa theory, etc), and I'll explain how the technique generalizes to certain families of higher genus curves/higher dimensional abelian varieties.   This is joint work with Alpöge and Bhargava.