Let $k \subset {\mathbb{C}}$ be a number field and $\mathcal{E}$ be an elliptic curve defined over $k(t)$ that is isomorphic to the generic fiber of an elliptic surface $\pi : \mathcal{S}_{\mathcal{E}} \rightarrow {\mathbb{P}}^1_k$. For any extension $\mathcal{K}\subseteq {\mathbb{C}}$ of $k$, the set $\mathcal{E}(\mathcal{K}(t))$ of $\mathcal{K}(t)$-rational points of $\mathcal{E}$ is known to be a finitely generated abelian group. The splitting field of $\mathcal{E}$ defined over $k(t)$ is the smallest finite extension $\mathcal{K} \subset {\mathbb{C}}$ of $k$ such that $\mathcal{E} ({\mathbb{C}} (t)) \cong \mathcal{E} (\mathcal{K}(t))$. In this paper, we consider the elliptic $K3$ surfaces defined over $k={\mathbb{Q}}$ with the generic fiber given by the Weierstrass equation $\mathcal{E}_n: \displaystyle y^2=x^3 + t^n + 1/t^n$, $1\le n\le 6$, and determine the splitting field $\mathcal{K}_n$, and find an explicit set of linearly independent generators for $\mathcal{E}_n (\mathcal{K_n}(t))$ for $1\le n \le 6$.
Soit $k \subset {\mathbb{C}}$ un corps de nombres et $\mathcal{E}$ une courbe elliptique définie sur $k(t)$ qui est isomorphe à la fibre générique d’une surface elliptique $\pi : \mathcal{S}_{\mathcal{E}} \rightarrow {\mathbb{P}}^1_k$. Pour tout sous-corps $\mathcal{K}\subseteq {\mathbb{C}}$ de $k$, l’ensemble $\mathcal{E}(\mathcal{K}(t))$ des points rationnels de $\mathcal{E}$ est un groupe abélien de type fini. Le corps de décomposition de $\mathcal{E}$ défini sur $k(t)$ est la plus petite extension finie $\mathcal{K} \subset {\mathbb{C}}$ de $k$ telle que $\mathcal{E} ({\mathbb{C}} (t)) \cong \mathcal{E} (\mathcal{K}(t))$. Dans cet article, nous considérons les surfaces elliptiques $K3$ définies sur $k={\mathbb{Q}}$ de fibre générique donnée par l’équation de Weierstrass $\mathcal{E}_n : \displaystyle y^2=x^3 + t^n + 1/t^n$, $1\le n\le 6$, déterminons le corps de décomposition $\mathcal{K}_n$ et trouvons un ensemble explicite de générateurs linéairement indépendants pour $\mathcal{E}_n (\mathcal{K_n}(t))$ pour $1 \le n \le 6$.
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Keywords: Elliptic surface, Mordell–Weil lattice, Splitting field
Sajad Salami 1 ; Arman Shamsi Zargar 2
CC-BY-ND 4.0
Sajad Salami; Arman Shamsi Zargar. Generators and splitting fields of certain elliptic $K3$ surfaces. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 2, pp. 317-339. doi: 10.5802/jtnb.1363
@article{JTNB_2026__38_2_317_0,
author = {Sajad Salami and Arman Shamsi Zargar},
title = {Generators and splitting fields of certain elliptic $K3$ surfaces},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {317--339},
year = {2026},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {38},
number = {2},
doi = {10.5802/jtnb.1363},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1363/}
}
TY - JOUR AU - Sajad Salami AU - Arman Shamsi Zargar TI - Generators and splitting fields of certain elliptic $K3$ surfaces JO - Journal de théorie des nombres de Bordeaux PY - 2026 SP - 317 EP - 339 VL - 38 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1363/ DO - 10.5802/jtnb.1363 LA - en ID - JTNB_2026__38_2_317_0 ER -
%0 Journal Article %A Sajad Salami %A Arman Shamsi Zargar %T Generators and splitting fields of certain elliptic $K3$ surfaces %J Journal de théorie des nombres de Bordeaux %D 2026 %P 317-339 %V 38 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1363/ %R 10.5802/jtnb.1363 %G en %F JTNB_2026__38_2_317_0
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