Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on a question of Shioda about integral sections

Jia-Li Mo

TL;DR

This work resolves Shioda's question on the number of integral sections for rational elliptic surfaces by combining the Mordell-Weil lattice framework with a detailed height-pairing analysis and Shioda's combinatorial multiplicity. Through a case-by-case treatment of the 74 Mordell-Weil types and two defining situations (S1,S2), the authors derive exact counts of integral sections and establish a key link between integrality and the narrow Mordell-Weil subgroup $E(K)^0$, supplemented by a dual-lattice bound to control multiplicities. The paper further connects these geometric counts to arithmetic by bounding the number of polynomial solutions to certain Weierstrass equations via the discriminant pattern $\Delta(t)$ and the topology of the elliptic fibration, with explicit examples showing the bounds are sharp. Overall, the results provide a complete classification of integral sections on rational elliptic surfaces and illustrate how discriminants constrain polynomial solutions in related Diophantine problems.

Abstract

We consider a rational elliptic surface with a relatively minimal fibration. We compute the number of integral sections in the above rational elliptic surface. As an application, we obtain an estimate of polynomial solutions of some equations.

A note on a question of Shioda about integral sections

TL;DR

This work resolves Shioda's question on the number of integral sections for rational elliptic surfaces by combining the Mordell-Weil lattice framework with a detailed height-pairing analysis and Shioda's combinatorial multiplicity. Through a case-by-case treatment of the 74 Mordell-Weil types and two defining situations (S1,S2), the authors derive exact counts of integral sections and establish a key link between integrality and the narrow Mordell-Weil subgroup , supplemented by a dual-lattice bound to control multiplicities. The paper further connects these geometric counts to arithmetic by bounding the number of polynomial solutions to certain Weierstrass equations via the discriminant pattern and the topology of the elliptic fibration, with explicit examples showing the bounds are sharp. Overall, the results provide a complete classification of integral sections on rational elliptic surfaces and illustrate how discriminants constrain polynomial solutions in related Diophantine problems.

Abstract

We consider a rational elliptic surface with a relatively minimal fibration. We compute the number of integral sections in the above rational elliptic surface. As an application, we obtain an estimate of polynomial solutions of some equations.
Paper Structure (5 sections, 7 theorems, 18 equations, 2 figures, 10 tables)

This paper contains 5 sections, 7 theorems, 18 equations, 2 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Acknowledgements:
  3. Definitions and terminology
  4. Proof of the Theorem 1
  5. Proof of Theorem 2

Key Result

Theorem 1

The structures of integral sections of every rational elliptic surface are given in table tab1:my_label--tab10:my_label. In these tables, the rows are arranged in the following order and we also give other indices, they descend with respect to:

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • proof