Belyi Lattes Maps

TL;DR

This paper classifies all Lattes maps that are Belyi morphisms, revealing conditions under which these maps form continuous families, especially in cases involving elliptic curves with complex multiplication.

Contribution

It explicitly determines when Lattes maps are Belyi morphisms, including the case of complex multiplication, and provides formulas for specific instances.

Findings

Lattes maps are Belyi morphisms iff the isogeny is multiplication by two in the generic case.

A continuous family of Belyi maps arises from certain Lattes maps.

Explicit formulas are given for Belyi morphisms with complex multiplication by a third root of unity.

Abstract

In this work, we determine all Lattes maps which are Belyi morphisms. It turns out that in the generic case, i.e. when the automorphism group is $\ZZ/2\ZZ$, the corresponding family of Lattes maps are Belyi morphisms if and only if the isogeny is multiplication by two. This family form a continuous family of Belyi maps. Elliptic curves with complex multiplication also determine a family over $\ZZ$ of Belyi morphisms. We give the explicit formulas for the first few Belyi morphisms when the curve has complex multiplication by 3rd root of unity.