2019/11/06 by Yang, Wenzhe
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1911.02608
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.