Apéry's irrationality proof, Beukers's modular forms and mirror symmetry

In this paper, we will apply the ideas from the mirror symmetry of Calabi-Yau threefolds to study the modular forms and one-parameter family of K3 surfaces found by Beukers and Peters, which provide enlightenment to the two mysterious sequences constructed by Apéry in his proof of the irrationality of $ζ(3)$. We will construct a fourth order differential operator and a prepotential from the canonical solutions of this differential operator. The third derivative of this prepotential with respect to the mirror map defines a Yukawa coupling that is a weight-4 modular form. The instanton expansion of this Yukawa coupling yields integral instanton numbers, which are also periodic with period 6.

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