Arithmetic Properties of Hypergeometric Mirror Maps and Dwork’s Congruences

Mirror maps are power series which occur in Mirror Symmetry as the inverse for composition of q ( z ) = exp ( f ( z ) ∕ g ( z ) ) $$q(z)=\exp (f(z)/g(z))$$ , called local q-coordinates, where f and g are particular solutions of the Picard–Fuchs differential equations associated with certain one-parameter families of Calabi–Yau varieties. In several cases, it has been observed that such power series have integral Taylor coefficients at the origin. In the case of hypergeometric equations, we discuss p-adic tools and techniques that enable one to prove a criterion for the integrality of the coefficients of mirror maps. This is a joint work with T. Rivoal and J. Roques. This note is an extended abstract of the talk given by the author in January 2017 at the conference “Hypergeometric motives and Calabi–Yau differential equations” in Creswick, Australia.