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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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