Abstract

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 on the Tate curve over Z[[q]]. It specializes to an equivalence, over Z, of the Fukaya category of the punctured torus with perfect complexes on the nodal Weierstrass curve y^2+xy=x3, and, over the punctured disc 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 my joint work with Tim Perutz.