Some Obstructions to Solvable Points on Higher Genus Curves
James Rawson
TL;DR
The paper investigates solvable points on curves of genus g ≥ 5 by linking the problem to the Bombieri-Lang conjecture and studying the geometry of quotient varieties C^n / G for transitive solvable G ≤ S_n. It establishes a transposition-based criterion: C^n / G is of general type whenever the genus g of C satisfies g > m+1, where m is the number of transpositions in G, and reduces the general case to symmetric powers via a normal subgroup, enabling a robust general-type conclusion when g is large. It further shows that, in the no-transpositions case, C^n / G is of general type for g ≥ 2, using a Riemann-Hurwitz-type analysis and construction of independent sections; the general case is completed by factoring through a product of symmetric powers and applying known results for Sym^n C. The fibre-type analysis ties potential solvable points to morphisms C → P^1 with Galois group G, and the Hilbert irreducibility framework yields concrete finiteness results in explicit examples (e.g., X_0(34)). Collectively, the work provides structural constraints on solvable points on high-genus curves and, under Bombieri-Lang, implies non-denseness of such points on the associated quotient varieties, refining our understanding of how arithmetic and geometry interact in this setting.
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.