Belyi Lattes Maps

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.