The Mordell-Weil lattice of an Inose surface arising from isogenous elliptic curves

TL;DR

This paper presents a method to determine rational sections of Inose surfaces derived from isogenous elliptic curves, with explicit examples for isogenies of degrees 5 and 6, advancing understanding of their Mordell-Weil lattices.

Contribution

It introduces a new method to find rational sections of Inose surfaces associated with isogenies between elliptic curves, including explicit examples for degrees 5 and 6.

Findings

Explicit bases for Mordell-Weil lattices of Inose surfaces from degree 5 and 6 isogenies

A general method to find rational sections corresponding to isogenies

Enhanced understanding of the structure of Inose surfaces and their Mordell-Weil groups

Abstract

An elliptic K3 surface having two $II^{*}$ fibers is called the Inose surface. In this paper, we give a method to find a rational section of an Inose surface corresponding to an isogeny of general degree between two elliptic curves. In particular, we show examples of bases of the Mordell-Weil lattices of Inose surfaces arising from isogenies of degrees 5 and 6.