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Moderate rank jumps on rational elliptic surfaces via construction of conics

Julie Desjardins

TL;DR

The paper addresses the problem of producing rational elliptic surfaces with fibers that exhibit rank jumps, aiming for ranks such as 3 or 4 across infinitely many fibers. It blends Walsh's two-section conic framework with LS20 base-change ideas to construct explicit parametric surfaces and genus-0 curves (bisections) that generate new independent sections on many fibers. The main contributions include explicit families (e.g., surface D) with proven or conjecturally infinite high-rank fibers, generalizations to higher ranks, and a suite of additional examples not previously covered, along with independence and non-torsion analyses. Together, these results provide a concrete recipe to generate infinite families of rational elliptic surfaces with elevated Mordell-Weil rank, advancing understanding of rank distribution in arithmetic geometry and offering tools for further exploration of rank jumps via conic base changes and genus-0 curves.

Abstract

This note is devoted to studying certain families of elliptic surfaces with infinitely many fibers with rank at least 3 or 4 revisiting and combining ideas from of Gary Walsh, Salgado and Loughran, and the author.

Moderate rank jumps on rational elliptic surfaces via construction of conics

TL;DR

The paper addresses the problem of producing rational elliptic surfaces with fibers that exhibit rank jumps, aiming for ranks such as 3 or 4 across infinitely many fibers. It blends Walsh's two-section conic framework with LS20 base-change ideas to construct explicit parametric surfaces and genus-0 curves (bisections) that generate new independent sections on many fibers. The main contributions include explicit families (e.g., surface D) with proven or conjecturally infinite high-rank fibers, generalizations to higher ranks, and a suite of additional examples not previously covered, along with independence and non-torsion analyses. Together, these results provide a concrete recipe to generate infinite families of rational elliptic surfaces with elevated Mordell-Weil rank, advancing understanding of rank distribution in arithmetic geometry and offering tools for further exploration of rank jumps via conic base changes and genus-0 curves.

Abstract

This note is devoted to studying certain families of elliptic surfaces with infinitely many fibers with rank at least 3 or 4 revisiting and combining ideas from of Gary Walsh, Salgado and Loughran, and the author.

Paper Structure

This paper contains 9 sections, 7 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Strategy
  4. Construct $S_1$ and $S_2$
  5. The sections $S_1$, $S_2$ are non-torsion
  6. Independance of $S_1$ and $S_2$ in the Mordell-Weil lattice
  7. Proof and an extension
  8. Search for high rank elliptic curves
  9. Similar examples

Key Result

Theorem 1.2

Let $s,w\in\mathbb{Q}$ be distinct numbers such that the conic $Y^2+(s+w)T^2=sw$ is solvable. Let $(t,m)$ be a solution. Then for a given $v\in\mathbb{Q}$, there is only finitely many exceptions to the fact that the elliptic curve given by the Weierstrass equation has rank at least 3, with the three following independant non-torsion points $[-t^2+vt,tm]$, $[vt+s,t^3]$, $[vt+w,t^3]$.

Theorems & Definitions (23)

  • Example 1.1
  • Theorem 1.2
  • Remark 1.3
  • Example 1.4
  • Remark 1.5
  • Example 1.6
  • Remark 1.7
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • ...and 13 more