Moderate rank jumps on rational elliptic surfaces via construction of conics

TL;DR

This paper investigates elliptic surfaces with infinitely many fibers of high rank, using a combination of existing ideas to construct conics that facilitate the analysis of rank jumps.

Contribution

It introduces a novel approach to constructing conics on rational elliptic surfaces, enabling the study of rank jumps in fibers with rank at least 3 or 4.

Findings

Identification of families with infinite fibers of high rank

Construction methods for conics on elliptic surfaces

Enhanced understanding of rank jumps in rational elliptic surfaces

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.