we explore 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 over the "punctured disc" spec z((q)), to an integral refinement of the known statement of homological mirror symmetry for the 2-torus. we will survey a general strategy of proof of homological mirror symmetry while carrying it out in the specific case of the 2-torus. in contrast to the abstract statement of our main result, the focus of the talk will be a concrete computation which we will express in more familiar terms. this is joint work with tim perutz.