Galois sections for abelian varieties over number fields

Mirela Ciperiani; Jakob Stix

doi:10.5802/jtnb.892

Mirela Ciperiani ¹; Jakob Stix ²

¹ Department of Mathematics The University of Texas at Austin 1 University Station C1200, Austin, Texas 78712, USA
² Institut für Mathematik Goethe–Universität Frankfurt Robert-Mayer-Straße 6–8, 60325 Frankfurt am Main, Germany

Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 47-52.

Abstract
Résumé

For an abelian variety $A$ over a number field $k$ , we discuss the space of sections of its fundamental group extension $π_{1} (A / k)$ . By analyzing the maximal divisible subgroup of $H^{1} (k, A)$ we show that the space of sections of $π_{1} (A / k)$ contains a copy of ${\hat{ℤ}}^{[k : ℚ] \cdot dim (A)}$ and is never in bijection with $A (k)$ . This is essentially a result about the structure of $H^{1} (k, T_{ℓ} (A))$ .

Pour une variété abélienne $A$ sur un corps de nombres $k$ , on considère l’espace des sections de l’extension $π_{1} (A / k)$ induite par le groupe fondamental. En étudiant le sous-groupe divisible maximal de $H^{1} (k, A)$ , on montre que l’espace des sections de $π_{1} (A / k)$ contient un sous-groupe isomorphe à ${\hat{ℤ}}^{[k : ℚ] \cdot dim (A)}$ et n’est jamais en bijection avec $A (k)$ . C’est essentiellement un résultat de structure pour $H^{1} (k, T_{ℓ} (A))$ .

MR

DOI: 10.5802/jtnb.892

Classification: 11G10, 11S25

Author's affiliations:

Mirela Ciperiani ¹; Jakob Stix ²

¹ Department of Mathematics The University of Texas at Austin 1 University Station C1200, Austin, Texas 78712, USA
² Institut für Mathematik Goethe–Universität Frankfurt Robert-Mayer-Straße 6–8, 60325 Frankfurt am Main, Germany

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     title = {Galois sections for abelian varieties over number fields},
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Mirela Ciperiani; Jakob Stix. Galois sections for abelian varieties over number fields. Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 47-52. doi : 10.5802/jtnb.892. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.892/

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