8-class ranks of imaginary quadratic fields and 4-Selmer groups of elliptic curves
Location
Take a and b to be integers that obey some technical conditions, and take H to be a large positive integer. If an integer d is chosen uniformly at random from [1, H], we prove that the resulting curve
E^d: dy^2 = x^3 + ax + b
has probability at most 1/2 + o(1) of having infinitely many rational points (x, y), partially resolving a conjecture of Goldfeld from 1979. To do this, we will find the distribution of 2^k-Selmer ranks in this family of curves for every k > 0. Using this framework, we will also find the distribution of the 2^k-class ranks of the imaginary quadratic fields for all k > 1.
We prove that the two-primary subgroups of the class groups of imaginary quadratic fields have the distribution predicted by the Cohen-Lenstra-Gerth heuristic. In this talk, we will detail our method for proving the 8-class rank portion of this theorem and will compare our approach to one that uses the governing fields predicted by Cohn and Lagraias. We will also connect this work to related questions on the 4-Selmer groups of elliptic curves in quadratic twist families.