Some Obstructions to Solvable Points on Higher Genus Curves
James Rawson1
1 Mathematics Institute, University of Warwick, Coventry United Kingdom
Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 1009-1019.

It is known that for a curve defined over ℚ of genus g≤4, there exists a point on the curve defined over a solvable extension of ℚ. We relate points on curves of genus g≥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 ℚ de genre g≤4 admet un point défini sur une extension résoluble de ℚ. Nous établissons un lien entre les points des courbes de genre g≥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.

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DOI : 10.5802/jtnb.1304
Classification : 11G30, 11G35, 14G05
Mots-clés : Higher genus curves, solvable points, quotient varieties, rational points, solvable morphisms

James Rawson 1

1 Mathematics Institute, University of Warwick, Coventry United Kingdom
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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James Rawson. Some Obstructions to Solvable Points on Higher Genus Curves. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 1009-1019. doi : 10.5802/jtnb.1304. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1304/

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