Some coverings defined over \(\mathbb{Q}\)
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.
ramified covering of \(\mathbb{P}^1-\{0,1,\infty\}\)
scientific article; zbMATH DE number 1038275
scientific article