Some Obstructions to Solvable Points on Higher Genus Curves

TL;DR

This paper explores the limitations of finding rational points on higher genus curves over solvable extensions, linking the problem to the Bombieri-Lang conjecture and the geometry of related varieties.

Contribution

It establishes a connection between points over solvable extensions on genus ≥5 curves and the general type of associated parameter varieties, highlighting obstructions.

Findings

Varieties parametrising points over solvable extensions are of general type.

Existence of certain subvarieties implies solvable morphisms from the curve.

Results relate to the Bombieri-Lang conjecture and higher genus curve rational points.

Abstract

It is known that for a curve defined over $\mathbb{Q}$ of genus $g \leq 4$, there exists a point on the curve defined over a solvable extension of $\mathbb{Q}$. We relate points on curves of genus $g \geq 5$ 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.