  This paper explores a refinement of homological mirror symmetry which relates
exact symplectic topology to arithmetic algebraic geometry. We establish a
derived equivalence of the Fukaya category of the 2-torus, relative to a
basepoint, with the category of perfect complexes of coherent sheaves on the
Tate curve over the "formal disc" Spec Z[[q]]. It specializes to a derived
equivalence, over Z, of the Fukaya category of the punctured torus with perfect
complexes on the curve y^2+xy=x^3 over Spec Z, the central fibre of the Tate
curve; and, over the "punctured disc" Spec Z((q)), to an integral refinement of
the known statement of homological mirror symmetry for the 2-torus. We also
prove that the wrapped Fukaya category of the punctured torus is
derived-equivalent over Z to bounded complexes of coherent sheaves on the
central fiber of the Tate curve.
