Integrality of Picard-Fuchs differential equations of Kobayashi geodesics and applications

Abstract

We prove that the holomorphic solutions of Picard-Fuchs differential equations associated with one-parameter families of abelian varieties with real multiplication admit power series expansions with $S$-integral coefficients at a maximal unipotent monodromy point. This extends classical integrality results for hypergeometric functions and Bouw-M\"oller's work on Teichm\"uller curves. The integral solutions are related to the non-ordinary locus of the modulo $p$ reduction of the family, whose cardinality we bound in terms of the Euler characteristic and Lyapunov exponents of the base curve. In some cases, the non-ordinary locus can be recovered by truncating the integral solutions, as in Igusa's classical observation for the Legendre family. We also establish $S$-integrality of expansions of modular forms at cusps in terms of a modular function for (not necessarily arithmetic) Fuchsian groups with modular embeddings, and deduce congruences. These results are applied in subsequent work to construct lifts of partial Hasse invariants for rational curves in Hilbert modular varieties.