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The splitting field and generators of the elliptic surface $Y^2=X^3 +t^{360} +1$

Sajad Salami

TL;DR

We determine the splitting field ${\mathcal{K}}$ for Shioda's elliptic surface ${\mathcal{E}}: Y^2 = X^3 + t^{360} + 1$ over ${\mathbb{Q}}(t)$, proving that ${\mathcal{K}}$ is the compositum of two high-degree polynomials and locating explicit $68$ generators for ${\mathcal{E}}({\mathcal{K}}(t))$. The method decomposes the Mordell-Weil lattice into the lattices of ten rational elliptic surfaces ${\mathcal{E}}_{a,b}$ and a $K3$ surface ${\mathcal{E}}'$, using birational maps, minimal polynomials, and height pairings, with verification via symbolic computation in Maple and PARI/GP. The authors confirm a maximal rank of $68$ over ${\mathbb{C}}(t)$ for this isotrivial $j=0$ surface and provide a complete computational construction of the splitting field and the 68 independent sections, organized through a detailed base-change framework. This work highlights the interplay between high-degree field extensions and Mordell-Weil group structure, delivering an exact, verified description of the arithmetic of a landmark elliptic surface.

Abstract

The splitting field of an elliptic surface $\mathcal{E}/\mathbb{Q}(t)$ is the smallest finite extension $\mathcal{K} \subset \mathbb{C}$ such that all $\mathbb{C}(t)$-rational points are defined over $\mathcal{K}(t)$. In this paper, we provide a symbolic algorithmic approach to determine the splitting field and a set of $68$ linearly independent generators for the Mordell--Weil lattice of Shioda's elliptic surface $Y^2=X^3 +t^{360} +1$. This surface is noted for having the largest known rank 68 for an elliptic curve over $\mathbb{C}(t)$. Our methodology utilizes the known decomposition of the Mordell-Weil Lattice of this surface into Lattices of ten rational elliptic surfaces and one $K3$ surface. We explicitly compute the defining polynomials of the splitting field, which reach degrees of 1728 and 5760, and verify the results via height pairing matrices and specialized symbolic software packages.

The splitting field and generators of the elliptic surface $Y^2=X^3 +t^{360} +1$

TL;DR

We determine the splitting field for Shioda's elliptic surface over , proving that is the compositum of two high-degree polynomials and locating explicit generators for . The method decomposes the Mordell-Weil lattice into the lattices of ten rational elliptic surfaces and a surface , using birational maps, minimal polynomials, and height pairings, with verification via symbolic computation in Maple and PARI/GP. The authors confirm a maximal rank of over for this isotrivial surface and provide a complete computational construction of the splitting field and the 68 independent sections, organized through a detailed base-change framework. This work highlights the interplay between high-degree field extensions and Mordell-Weil group structure, delivering an exact, verified description of the arithmetic of a landmark elliptic surface.

Abstract

The splitting field of an elliptic surface is the smallest finite extension such that all -rational points are defined over . In this paper, we provide a symbolic algorithmic approach to determine the splitting field and a set of linearly independent generators for the Mordell--Weil lattice of Shioda's elliptic surface . This surface is noted for having the largest known rank 68 for an elliptic curve over . Our methodology utilizes the known decomposition of the Mordell-Weil Lattice of this surface into Lattices of ten rational elliptic surfaces and one surface. We explicitly compute the defining polynomials of the splitting field, which reach degrees of 1728 and 5760, and verify the results via height pairing matrices and specialized symbolic software packages.
Paper Structure (16 sections, 15 theorems, 70 equations, 4 tables)

This paper contains 16 sections, 15 theorems, 70 equations, 4 tables.

Table of Contents

  1. Introduction and the main result
  2. Sketch of the proof of Theorem \ref{['main0']}
  3. Some results on the elliptic surfaces ${\mathcal{E}}_{a,b}$
  4. Birational map between ${\mathcal{E}}_{a,b}$
  5. Singular Fibers and Mordell-Weil Lattice of Rational ${\mathcal{E}}_{a,b}$
  6. Transformation of points of ${\mathcal{E}}_{a,b}$ on ${\mathcal{E}}$
  7. The splitting field and generators of ${\mathcal{E}}_{2,1}$ and ${\mathcal{E}}_{3,1}$
  8. The splitting field and generators of ${\mathcal{E}}_{1,2}$ and ${\mathcal{E}}_{3,2}$
  9. The splitting field and generators of ${\mathcal{E}}_{2,2}$
  10. The splitting field and generators of ${\mathcal{E}}_{1, 3}$ and ${\mathcal{E}}_{2, 3}$
  11. The splitting field and generators of ${\mathcal{E}}_{1, 4}$
  12. The splitting field and generators of ${\mathcal{E}}_{0,5}$ and ${\mathcal{E}}_{1,5}$
  13. The splitting field and generators of the $K3$ surface ${\mathcal{E}}'$
  14. Computational proof of Theorem \ref{['main0']}
  15. Computational Construction of the $68 \times 68$ Gram Matrix
  16. ...and 1 more sections

Key Result

Theorem 1.1

The splitting field ${\mathcal{K}}$ of the Shioda's elliptic curve shi-eq1 is the compositum field of two number fields defined by two polynomials of degree 1728 and 5760, which contains The Mordell-Weil group ${\mathcal{E}}({\mathcal{K}}(t))$ is generated by $P_1,\cdots, P_{68}$, as listed below: The Gram matrix $M_{68}$ of these 68 points is a diagonal matrix with 11 blocks M68 with

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 4.1
  • ...and 14 more