Some Obstructions to Solvable Points on Higher Genus Curves

Rawson, James

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 1009-1019

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.

On sait que toute courbe algébrique sur $\mathbb{Q}$ de genre $g\le 4$ admet un point défini sur une extension résoluble de $\mathbb{Q}$. Nous établissons un lien entre les points des courbes de genre $g\ge 5$ définis sur les extensions résolubles et la conjecture de Bombieri–Lang. Plus précisément, nous montrons que les variétés paramétrant les points définis sur les extensions de groupe de Galois résoluble fixé sont de type général. En outre, nous montrons que l’existence de certaines sous-variétés de ces variétés implique l’existence de morphismes résolubles définies sur la courbe.

Classification : 11G30, 11G35, 14G05
Keywords: Higher genus curves, solvable points, quotient varieties, rational points, solvable morphisms

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