Some Obstructions to Solvable Points on Higher Genus Curves

Abstract

It is known that for a curve defined over $\mathbb{Q}$ of genus $g\le 4$, there exists a point on the curve defined over a solvable extension of $\mathbb{Q}$. We relate points on curves of genus $g\ge 5$ over solvable extensions to the Bombieri-Lang conjecture. Specifically, we show that varieties parametrising points defined over extensions with a fixed solvable Galois group are of general type. Moreover, we show the existence of certain subvarieties in these varieties imply the existence of solvable morphisms from the curve.

Version appearing in the Bordeaux Journal of Number Theory