Mordell-Weil lattice of Inose's Elliptic $K3$ surface arising from the product of 3-isogenous elliptic curves

TL;DR

This paper develops a method to explicitly determine the Mordell-Weil lattice generators for certain elliptic K3 surfaces derived from 3-isogenous elliptic curves, including explicit bases for specific singular cases.

Contribution

It introduces a new approach to compute Mordell-Weil lattices for K3 surfaces from 3-isogenous elliptic curves, providing explicit bases for particular singular cases.

Findings

Explicit generators for Mordell-Weil lattices when elliptic curves are 3-isogenous.

Bases for Mordell-Weil lattices of specific singular K3 surfaces.

Method applicable to elliptic surfaces with two $ ext{II}^*$ fibers.

Abstract

From the product of two elliptic curves, Shioda and Inose constructed an elliptic $K3$ surface having two $\mathrm{II}^*$ fibers. Its Mordell-Weil lattice structure depends on the morphisms between the two elliptic curves. In this paper, we give a method of writing down generators of the Mordell-Weil lattice of such elliptic surfaces when two elliptic curves are $3$-isogenous. In particular, we obtain a basis of the Mordell-Weil lattice for the singular $K3$ surfaces $X_{[3,3,3]}$, $X_{[3,2,3]}$ and $X_{[3,0,3]}$.