🤖 AI Summary
This work addresses the computation of a homology basis for complex elliptic surfaces over ℙ¹, enabling period integrals and reconstruction of key algebraic invariants. We introduce the first systematic semi-numerical algorithm, implemented as an end-to-end framework in SageMath, integrating high-precision numerical integration, symbolic computation, and algebraic geometry theory. Our contribution is threefold: (i) we pioneer the application of semi-numerical techniques to construct homology bases for elliptic surfaces—overcoming limitations of purely symbolic or purely numerical approaches; (ii) we stably recover the Néron–Severi lattice, transcendental lattice, Mordell–Weil group, and its associated lattice structure; and (iii) the method significantly enhances the feasibility of period computations, providing a new computational paradigm for the classification of moduli spaces of elliptic surfaces and effective arithmetic-geometric calculations.
📝 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.