香蕉视频

Title: Sums of two cubes.

Abstract:聽I'll give an overview of recent work with Alpoge-Bhargava showing that at least 10/21 (resp. 1/6) of integers are (resp. are not) a sum of two rational cubes. To prove this, we first show that the average size of the 2-Selmer group in any cubic twist family of elliptic curves is 3, by reducing to a certain counting problem: count integral G-orbits in a G-invariant quadric inside V, where G = SL_2^2 and V is the space of pairs of binary cubic forms. To perform the lattice point count, we combine tools from geometry of numbers and the circle method. 聽My talk will focus on the more algebraic aspects of the problem, e.g.: how one reduces to the counting problem, and how one deduces the results about sums of two cubes from the Selmer average result. If I have time, I'll explain a result for cubic twist families of higher dimensional abelian varieties as well.