Solvable points on genus one curves

Abstract

A genus one curve defined over $\mathbb{Q}$ which has points over $\mathbb{Q}_{p}$ for all primes $p$ may not have a rational point. It is natural to study the classes of $\mathbb{Q}$-extensions over which all such curves obtain a global point. In this article, we show that every such genus one curve with semistable Jacobian has a point defined over a solvable extension of $\mathbb{Q}$