Arithmetic mirror symmetry for genus 1 curves with n marked points

Abstract

We establish a \({\mathbb {Z}}[[t_1,\ldots , t_n]]\)-linear derived equivalence between the relative Fukaya category of the 2-torus with n distinct marked points and the derived category of perfect complexes on the n-Tate curve. Specialising to \(t_1= \cdots =t_n=0\) gives a \({\mathbb {Z}}\)-linear derived equivalence between the Fukaya category of the n-punctured torus and the derived category of perfect complexes on the standard (Neron) n-gon. We prove that this equivalence extends to a \({\mathbb {Z}}\)-linear derived equivalence between the wrapped Fukaya category of the n-punctured torus and the derived category of coherent sheaves on the standard n-gon. The corresponding results for \(n=1\) were established in Lekili and Perutz (Arithmetic mirror symmetry for the 2-torus (preprint) arXiv:1211.4632, 2012).