Generalized Goldfeld’s conjecture for quadratic twist families

Establish the generalized Goldfeld’s conjecture for the quadratic twist family t y^2 = f(x) over a number field K by proving the predicted precise probabilities for Mordell–Weil rank 0 and rank 1 among twists, as determined by the distribution of root numbers.

Background

For the family of curves given by t y^2 = f(x), the authors note that one can compute the distribution of root numbers and thereby formulate a generalized version of Goldfeld’s conjecture predicting explicit rank 0 and rank 1 proportions. While recent breakthroughs control 2^∞-Selmer distributions, translating these into unconditional rank results remains out of reach in general.

Their main theorem secures the existence of rank 1 twists in a generic full rational 2-torsion setting, but the full distributional statement of Goldfeld’s conjecture for these families remains a major target.