  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.
