The platonic elliptic surfaces
Nutsa Gegelia, Duco van Straten
Source 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 Apéry 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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