Arithmetic mirror symmetry for the 2-torus

TL;DR

This paper establishes a refined form of homological mirror symmetry for the 2-torus, relating symplectic topology to arithmetic algebraic geometry through derived equivalences involving the Tate curve.

Contribution

It proves a derived equivalence between the Fukaya category of the 2-torus and perfect complexes on the Tate curve over Z, extending mirror symmetry to an arithmetic setting.

Findings

Derived equivalence between Fukaya category and perfect complexes on Tate curve

Specialization to classical mirror symmetry for punctured torus

Wrapped Fukaya category equivalent to coherent sheaves on Tate curve's central fiber

Abstract

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.