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

TL;DR

The paper presents a semi-numerical algorithm to compute the full homology lattice $H_2(S)$ and the holomorphic period map for complex elliptic surfaces $f:S\to \mathbb{P}^1$, enabling heuristic recovery of the Néron–Severi lattice and Mordell–Weil data. It builds on monodromy and extension theory, extends Lefschetz-fibration methods via morsification to general elliptic surfaces, and uses Picard–Fuchs equations together with LLL to extract lattice information. The approach is implemented in SageMath (lefschetz-family) and demonstrated on a high-rank Mordell–Weil example, illustrating practical computation for invariants central to number theory, algebraic geometry, and mathematical physics. While powerful, the NS/MW recovery is heuristic and relies on numerical period computations, with caveats about potential missed or spurious relations. Overall, the work provides a scalable computational toolkit linking topology, period integrals, and arithmetic of elliptic surfaces with tangible applications to areas such as Feynman integrals and mirror symmetry.

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 Néron-Severi lattice, the transcendental lattice, the Mordell-Weil group and the Mordell-Weil lattice. This algorithm comes with a SageMath implementation.

Figures (2)

• Figure 1: Distinguished bases and thimbles.
• Figure 2: The morsification of a neighbourhood of a single critical value. Left: A neighbourhood $V$ of a single critical value of the elliptic fibration $f$, along with a chosen basepoint $b$. The homotopy group $\pi_1(V\setminus\{c\}, b)$ is generated by the counterclockwise loop $\ell_\infty$. Right: The neighbourhood after morsification. The critical fibre has split into five Lefschetz fibres. The homotopy group $\pi_1(V\setminus \Sigma)$ is generated by the $5$ counterclockwise loops $\ell_1, \dots, \ell_5$. Thus $H_2(f^{-1}(V), F_b)$ has rank $5$ --- it follows from Table \ref{['tab:KodairaClassification']} that the original singular fibre was of type $I_5$. Furthermore we see that $\tau_{\ell_\infty} = \tau_{\ell_5\dots \ell_1}$.

Theorems & Definitions (40)

• Definition 1
• Definition 2
• Lemma 1
• proof
• Definition 3
• Theorem 2
• Definition 4
• Lemma 3: lamotke
• Lemma 4
• proof