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

Wenzhe Yang

Published 2019-11-06Version 1

In this paper, we will study the connections between Ap\'ery's proof of $\zeta(3)$'s irrationality and the mirror symmetry of Calabi-Yau threefolds. From the mysterious sequences in Ap\'ery's proof, we will construct a fourth order Picard-Fuchs equation that has a large complex structure limit. While the mirror map is the modular form of $\Gamma_1(6)$ found by Beukers. We will show how mirror symmetry provides further enlightening explanations to Beukers' and many others' enlightening explanations to Ap\'ery's mysterious proof.