Ap\'ery's irrationality proof, Beukers's modular forms and mirror symmetry

TL;DR

This paper explores the connection between Apéry's irrationality proof of zeta(3), Beukers's modular forms, and mirror symmetry of Calabi-Yau threefolds, revealing new insights into modular forms and instanton numbers.

Contribution

It introduces a novel differential operator and prepotential linked to mirror symmetry, connecting Apéry's sequences with modular forms and instanton expansions.

Findings

Constructed a fourth order differential operator.

Derived a Yukawa coupling as a weight-4 modular form.

Obtained integral instanton numbers with period 6.

Abstract

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 Ap\'ery in his proof of the irrationality of $\zeta(3)$. 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.