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Generators and splitting fields of certain elliptic K3 surfaces

Sajad Salami, Arman Shamsi Zargar

TL;DR

This work advances the explicit determination of splitting fields for elliptic K3 surfaces defined by y^2 = x^3 + t^n + t^{-n} over ℚ(t) for n=1,…,6 and provides fully explicit independent generators for the Mordell–Weil groups E_n(Κ_n(t)). Building on Shioda’s Mordell–Weil lattice framework, the authors decompose the problem through rational elliptic surfaces E'_n(s), compute fundamental polynomials to obtain splitting fields Κ'_n, and then form Κ_n by adjoining suitable roots of unity; they transform generator data back to the original t-parameter. The results yield ranks r_n = 0,4,8,12,16,16 for n=1,…,6, and give concrete Gram matrices with unimodular determinants in each case, together with explicit coordinates for 16- or fewer-point generating sets. The paper combines theoretical lattice analysis with algorithmic computations (Maple, PARI/GP, SageMath) to produce fully explicit splitting fields and generators, enabling further study of arithmetic and geometric properties of these elliptic K3 surfaces. Applications include understanding Galois actions on Mordell–Weil groups and providing templates for similar computations in broader families of K3 elliptic surfaces.

Abstract

Let $k \subset {\mathbb C}$ be a number field and ${\mathcal E}$ be an elliptic curve defined over $k(t)$, the rational function field of the projective line ${\mathbb P}^1_k$, is isomorphic to the generic fiber of an elliptic surface $π:= \Sc_\Ee \rightarrow {\mathbb P}^1_k$. For any subfield ${\mathcal K}\subseteq {\mathbb C}$ 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 ${\mathcal E} ({\mathbb C} (t)) \iso {\mathcal E} ({\mathcal K}(t))$. In this paper, we consider the elliptic $K3$ surfaces defined over $k={\mathbb Q}$ with the generic fiber given by the Weierstrass equation ${\mathcal E}_n: \displaystyle y^2=x^3 + t^n + 1/t^n$, $1\leq n\leq 6$, and determine the splitting field ${\mathcal K}_n$, and find an explicit set of independent generators for ${\mathcal E}_n ({\mathcal K_n}(t))$ for $1\leq n \leq 6$.

Generators and splitting fields of certain elliptic K3 surfaces

TL;DR

This work advances the explicit determination of splitting fields for elliptic K3 surfaces defined by y^2 = x^3 + t^n + t^{-n} over ℚ(t) for n=1,…,6 and provides fully explicit independent generators for the Mordell–Weil groups E_n(Κ_n(t)). Building on Shioda’s Mordell–Weil lattice framework, the authors decompose the problem through rational elliptic surfaces E'_n(s), compute fundamental polynomials to obtain splitting fields Κ'_n, and then form Κ_n by adjoining suitable roots of unity; they transform generator data back to the original t-parameter. The results yield ranks r_n = 0,4,8,12,16,16 for n=1,…,6, and give concrete Gram matrices with unimodular determinants in each case, together with explicit coordinates for 16- or fewer-point generating sets. The paper combines theoretical lattice analysis with algorithmic computations (Maple, PARI/GP, SageMath) to produce fully explicit splitting fields and generators, enabling further study of arithmetic and geometric properties of these elliptic K3 surfaces. Applications include understanding Galois actions on Mordell–Weil groups and providing templates for similar computations in broader families of K3 elliptic surfaces.

Abstract

Let be a number field and be an elliptic curve defined over , the rational function field of the projective line , is isomorphic to the generic fiber of an elliptic surface . For any subfield of , the set of -rational points of is known to be a finitely generated abelian group. The splitting field of defined over is the smallest finite extension of such that . In this paper, we consider the elliptic surfaces defined over with the generic fiber given by the Weierstrass equation , , and determine the splitting field , and find an explicit set of independent generators for for .
Paper Structure (15 sections, 9 theorems, 82 equations, 3 tables)

This paper contains 15 sections, 9 theorems, 82 equations, 3 tables.

Table of Contents

  1. Introduction and main results
  2. Shioda's results on Mordell--Weil lattice of ${\mathcal{E}}_n$
  3. An algorithmic approach to the proof of the theorems
  4. The cases ${\mathcal{E}}_2$ and ${\mathcal{E}}_3$
  5. Proof of Theorem \ref{['main-0']}
  6. Proof of Theorem \ref{['main-a']}
  7. The case of ${\mathcal{E}}_4$
  8. Proof of Theorem \ref{['main-b1']}
  9. Proof of Theorem \ref{['main-b']}
  10. The case of ${\mathcal{E}}_5$
  11. Proof of Theorem \ref{['main-c1']}
  12. Proof of Theorem \ref{['main-c']}
  13. The case of ${\mathcal{E}}_6$
  14. Proof of Theorem \ref{['main-d1']}
  15. Proof of Theorem \ref{['main-d']}

Key Result

Theorem 1.1

The Mordell--Weil lattice ${\mathcal{E}}_2({\mathbb C}(t))$ is isomorphic to ${\mathcal{E}}_2({\mathcal{K}}_2(t))$ with $r_2=4$, where ${\mathcal{K}}_2= {\mathbb Q}(\zeta_3, 2^{\frac{1}{3}})$ of degree 6 with a minimal defining polynomial $g_2(x)=x^6+ 108.$ Moreover, a set of linearly independent ge

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Theorem 5.1
  • Theorem 6.1
  • Theorem 7.1