The platonic elliptic surfaces

Nutsa Gegelia; Duco van Straten

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arXiv · 2608.25644

The platonic elliptic surfaces

Nutsa Gegelia·Duco van Straten

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Abstract

We construct certain special rational elliptic surfaces with four reduced singular fibres that are naturally attached to the platonic solids and belong to a group of surfaces complementing the well-known semi-stable Beauville surfaces. We describe the basic algebraic geometric properties of these surfaces, their associated Picard-Fuchs operators, integer sequences, Laurent polynomial representations and Ap\'ery constants. On the arithmetic side, we discover the need for a refinement of the Dwork-expansion used to find the Euler factors of the fibres of these fibrations. This is explained by comparing the $q$-coordinates coming from the elliptic curve and the $q$-coordinate of the Picard-Fuchs equation and is also directly reflected in the shape of the monodromy matrices in the Frobenius basis.

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Nutsa Gegelia, Duco van Straten. 2026-08-26. The platonic elliptic surfaces. https://arxiv.org/abs/2608.25644

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