Integrality of mirror maps and arithmetic homological mirror symmetry for Greene--Plesser mirrors

Abstract

We prove the `integrality of Taylor coefficients of mirror maps' conjecture for Greene--Plesser mirror pairs as a natural byproduct of an arithmetic refinement of homological mirror symmetry. We also prove homological mirror symmetry for Greene--Plesser mirror pairs in all characteristics such that the B-side family has good reduction, generalizing work of the fifth author and Smith over the complex numbers. A key technical ingredient is a new versality argument which allows us to work throughout over a Novikov-type ring with integer coefficients.