A semi-numerical algorithm for the homology lattice and periods of complex elliptic surfaces over the projective line

Abstract

We provide an algorithm for computing an effective basis of homology of elliptic surfaces over the complex projective line on which integration of periods can be carried out. This allows the heuristic recovery of several algebraic invariants of the surface, notably the N\'eron-Severi lattice, the transcendental lattice, the Mordell-Weil group and the Mordell-Weil lattice. This algorithm comes with a SageMath implementation.