Arithmetic mirror symmetry for genus 1 curves with $n$ marked points

TL;DR

This paper establishes a deep connection between the Fukaya category of a punctured torus and the derived category of perfect complexes on the n-Tate curve, revealing mirror symmetry phenomena for genus 1 curves with marked points.

Contribution

It constructs a new derived equivalence linking the Fukaya category of the punctured torus with the derived category of perfect complexes on the n-Tate curve, extending to wrapped Fukaya categories.

Findings

Derived equivalence between Fukaya category and perfect complexes on n-Tate curve

Extension of equivalence to wrapped Fukaya category and coherent sheaves

Specialization yields equivalence for standard n-gon and punctured torus

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 $t_1= \ldots =t_n=0$ 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 (N\'eron) $n$-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.