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Belyĭ uniformization of elliptic curves. (English) Zbl 0868.14019
Belyĭ’s theorem implies that a Riemann surface $$X$$ represents a curve defined over a number field if and only if it can be expressed as $$U/ \Gamma$$ where $$U$$ is simply-connected and $$\Gamma$$ is a subgroup of finite index in a triangle group. We consider the case when $$X$$ has genus 1 and ask for which curves and number fields can $$\Gamma$$ be chosen to be a lattice. As an application we give examples of Galois actions on Grothendieck dessins.

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 14L30 Group actions on varieties or schemes (quotients) 11G05 Elliptic curves over global fields 14H52 Elliptic curves 30F10 Compact Riemann surfaces and uniformization
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