On Grothendieck’s section conjecture for curves of index $1$
Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 295-316

Each curve for which Grothendieck’s section conjecture has been proved has no rational points, and additionally it has index different from $1$. We provide many new examples of curves satisfying the conjecture; in particular, we prove that examples of index $1$ are very common.

Given an odd prime $p$, we prove that every hyperbolic curve with a faithful action of a non-cyclic $p$-group has a twisted form of index $1$ which satisfies Grothendieck’s section conjecture. Furthermore, we prove that for every hyperbolic curve $S$ over a field $k$ finitely generated over $\mathbb{Q}$ there exists a finite extension $K/k$ and a finite étale cover $C \rightarrow S_{K}$ such that $C$ satisfies the conjecture.

Dans tous les cas où la conjecture de la section de Grothendieck a été démontrée, la courbe ne possède aucun point rationnel et, de plus, son indice est différent de $1$. Nous fournissons de nombreux nouveaux exemples de courbes satisfaisant la conjecture ; en particulier, nous démontrons que les exemples d’indice $1$ sont très fréquents.

Étant donné un nombre premier impair $p$, nous démontrons que toute courbe hyperbolique munie d’une action fidèle d’un $p$-groupe non cyclique admet une forme tordue d’indice $1$ qui satisfait la conjecture de la section de Grothendieck. De plus, nous montrons que, pour toute courbe hyperbolique $S$ définie sur un corps $k$ de type fini sur $\mathbb{Q}$, il existe une extension finie $K/k$ et un revêtement étale fini $C \rightarrow S_K$ tels que $C$ vérifie la conjecture.

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DOI : 10.5802/jtnb.1362
Classification : 11G30, 14H25, 14H30, 14G05
Keywords: Section conjecture, index of a curve
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Giulio Bresciani. On Grothendieck’s section conjecture for curves of index $1$. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 295-316. doi: 10.5802/jtnb.1362
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