On cyclic coverings of the torus

TL;DR

This paper investigates cyclic covering tori, characterizing their parameters arithmetically, analyzing their symmetry under SL(2,Z), and counting their occurrences both exactly and asymptotically.

Contribution

It provides a natural parametrization of cyclic covering tori, establishes their arithmetic characterization, and analyzes their symmetry and enumeration under SL(2,Z).

Findings

Cyclic covering tori are characterized by specific arithmetic conditions.

All n-tuple cyclic covers belong to a single SL(2,Z)-orbit.

The paper provides exact and asymptotic counts of cyclic covers.

Abstract

We study tori which are cyclic covers of the standard torus, that is, the deck transformation group of the covering map is cyclic. These covering tori can be parametrized in a natural way and we show that being cyclic is equivalent to certain arithmetic condition on these parameters. There is a natural $\mathrm{SL}(2,\mathbb{Z})$-action on covering tori and introducing a complete numeric $\mathrm{SL}(2,\mathbb{Z})$-invariant we show that, for $n\in\mathbb{N}$, all $n$-tuple cyclic covers are in the same $\mathrm{SL}(2,\mathbb{Z})$-orbit. We show that cyclic covers are irreducible in a precise sense and we give the exact and asymptotic number of these covers.