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Some coverings defined over \(\mathbb{Q}\). (Quelques revêtements définis sur \(\mathbb{Q}\).) (French) Zbl 0946.14015

This paper provides new examples that make explicit the Grothendieck correspondence between dessins d’enfants and Belyi pairs in some special cases. All examples are curves of genus 0 defined over \(\mathbb{Q}\), more precisely plane quadrics \(C_{m,n,p,q}\) depending on integer parameters \(m,n,p\) and \(q\), together with Belyi functions \(\lambda_{m,n,p,q}: C_{m,n,p,q} \to\mathbb{P}^1\). The author presents the corresponding (rather involved!) dessins by interpreting his curves as certain Hurwitz spaces and calculating the monodromy of the associated covering. He also studies in some detail the reduction of a plane quadric with integer coefficients at various primes; in particular he proves criteria for good and potentially good reduction of such a curve, and applies them to his examples.

MSC:

14H30 Coverings of curves, fundamental group
14G25 Global ground fields in algebraic geometry
14E20 Coverings in algebraic geometry
14H25 Arithmetic ground fields for curves
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References:

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