Transcendental lattice of an extremal elliptic surface

TL;DR

This paper presents an algorithm to compute the transcendental lattice and Mordell--Weil group of extremal elliptic surfaces, demonstrated through examples involving large series of such surfaces.

Contribution

The authors introduce a novel algorithm for calculating key invariants of extremal elliptic surfaces, including the transcendental lattice and Mordell--Weil group.

Findings

Successfully computed lattices for four large series of extremal elliptic surfaces.

Demonstrated the effectiveness of the algorithm on complex examples.

Provided new insights into the structure of extremal elliptic surfaces.

Abstract

We develop an algorithm computing the transcendental lattice and the Mordell--Weil group of an extremal elliptic surface. As an example, we compute the lattices of four exponentially large series of surfaces