Abelian geometric fundamental groups for curves over a $p$-adic field

Evangelia Gazaki; Toshiro Hiranouchi

doi:10.5802/jtnb.1269

Abelian geometric fundamental groups for curves over a

p

-adic field

Evangelia Gazaki ¹ ; Toshiro Hiranouchi ²

¹ Department of Mathematics, University of Virginia 221 Kerchof Hall, 141 Cabell Dr., Charlottesville, VA, 22904, USA
² Department of Basic Sciences, Graduate School of Engineering, Kyushu Institute of Technology 1-1 Sensui-cho, Tobata-ku, Kitakyushu-shi, Fukuoka 804-8550, Japan

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 905-946.

Résumé
Abstract

Pour une courbe $X$ sur un corps $p$ -adique $k$ , nous étudions le groupe fondamental géométrique abélien $π_{1}^{ab} {(X)}^{geo}$ de $X$ en utilisant la théorie du corps de classes de $X$ due à S. Bloch et S. Saito. En particulier, nous étudions un sous-groupe de $π_{1}^{ab} {(X)}^{geo}$ qui classifie les revêtements géométriques et abéliens de $X$ admettant une ramification au-dessus de la fibre spéciale du modèle de $X$ . En supposant que $X$ a un point rationnel sur $k$ , $X$ a bonne réduction et sa jacobienne a bonne réduction ordinaire, nous donnons un encadrement de ce sous-groupe de $π_{1}^{ab} {(X)}^{geo}$ .

For a curve $X$ over a $p$ -adic field $k$ , using the class field theory of $X$ due to S. Bloch and S. Saito we study the abelian geometric fundamental group $π_{1}^{ab} {(X)}^{geo}$ of $X$ . In particular, we investigate a subgroup of $π_{1}^{ab} {(X)}^{geo}$ which classifies the geometric and abelian coverings of $X$ which allow possible ramification over the special fiber of the model of $X$ . Under the assumptions that $X$ has a $k$ -rational point, $X$ has good reduction and its Jacobian variety has good ordinary reduction, we give some upper and lower bounds of this subgroup of $π_{1}^{ab} {(X)}^{geo}$ .

Reçu le : 2022-09-09
Accepté le : 2023-04-28
Publié le : 2024-03-04

DOI : 10.5802/jtnb.1269

Classification : 11G45, 11G10, 11G20
Mots clés : Class field theory, Fundamental groups, and Elliptic curves

Affiliations des auteurs :

Evangelia Gazaki ¹ ; Toshiro Hiranouchi ²

¹ Department of Mathematics, University of Virginia 221 Kerchof Hall, 141 Cabell Dr., Charlottesville, VA, 22904, USA
² Department of Basic Sciences, Graduate School of Engineering, Kyushu Institute of Technology 1-1 Sensui-cho, Tobata-ku, Kitakyushu-shi, Fukuoka 804-8550, Japan

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Abelian geometric fundamental groups for curves over a $p$-adic field},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Evangelia Gazaki; Toshiro Hiranouchi. Abelian geometric fundamental groups for curves over a $p$-adic field. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 905-946. doi : 10.5802/jtnb.1269. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1269/

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