Belyĭ functions, hypermaps and Galois groups.

*(English)*Zbl 0853.14017This is a survey article based on ideas of Grothendieck which provide geometric and combinatorial ways of studying the Galois theory of algebraic numbers. We report on some recent results in the field, and point out the relationship with the theory of hypermaps introduced by R. Cori in 1975.

A connected Riemann surface \(X\) is compact if and only if it is the Riemann surface of \(\Phi (x, y) =0\) for some irreducible polynomial \(\Phi\in \mathbb{C} [x, y]\); it is said to be defined over a subfield \(K\) of \(\mathbb{C}\) if \(\Phi\in K[x, y]\). Belyi’s theorem states that \(X\) is defined over the field \(\overline {\mathbb{Q}}\) of algebraic numbers if and only if there is a Belyi function \(\beta: X\to \Sigma\), that is, a meromorphic function from \(X\) to the Riemann sphere \(\Sigma= P^1 (\mathbb{C})\) with all its critical values in \(\{0, 1, \infty\}\). Such a covering \(\beta\) is determined by three monodromy permutations \(g_0\), \(g_1\) and \(g_\infty\) which show how the sheets are permuted by continuation around 0, 1 and \(\infty\); these generate a transitive group, and satisfy \(g_0 g_1 g_\infty =1\). One can always choose \(\beta\) so that \(g^2_1 =1\), in which case these permutations correspond to a map (a 2-cell imbedding of a graph) on the surface \(X\); this gives the correspondence, first pointed out by Grothendieck, between algebraic curves over \(\overline {\mathbb{Q}}\) and maps, or dessins d’enfants as he called them.

A general Belyi function \(\beta\) (without the restriction on \(g_1\)) similarly corresponds to a hypermap (or hypergraph imbedding) on \(X\), and in this survey we show how these give a more natural combinatorial representation of Belyi functions. In either case, the natural action of the absolute Galois group \(\mathbb{G}= \text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})\) of \(\overline {\mathbb{Q}}\) on Belyi pairs \((X, \beta)\) induces a faithful action of \(\mathbb{G}\) on maps or on hypermaps, thus giving an alternative insight into this important and complicated group. We explain how these topics are closely related to certain congruence subgroups of the modular group and to elliptic curves.

A connected Riemann surface \(X\) is compact if and only if it is the Riemann surface of \(\Phi (x, y) =0\) for some irreducible polynomial \(\Phi\in \mathbb{C} [x, y]\); it is said to be defined over a subfield \(K\) of \(\mathbb{C}\) if \(\Phi\in K[x, y]\). Belyi’s theorem states that \(X\) is defined over the field \(\overline {\mathbb{Q}}\) of algebraic numbers if and only if there is a Belyi function \(\beta: X\to \Sigma\), that is, a meromorphic function from \(X\) to the Riemann sphere \(\Sigma= P^1 (\mathbb{C})\) with all its critical values in \(\{0, 1, \infty\}\). Such a covering \(\beta\) is determined by three monodromy permutations \(g_0\), \(g_1\) and \(g_\infty\) which show how the sheets are permuted by continuation around 0, 1 and \(\infty\); these generate a transitive group, and satisfy \(g_0 g_1 g_\infty =1\). One can always choose \(\beta\) so that \(g^2_1 =1\), in which case these permutations correspond to a map (a 2-cell imbedding of a graph) on the surface \(X\); this gives the correspondence, first pointed out by Grothendieck, between algebraic curves over \(\overline {\mathbb{Q}}\) and maps, or dessins d’enfants as he called them.

A general Belyi function \(\beta\) (without the restriction on \(g_1\)) similarly corresponds to a hypermap (or hypergraph imbedding) on \(X\), and in this survey we show how these give a more natural combinatorial representation of Belyi functions. In either case, the natural action of the absolute Galois group \(\mathbb{G}= \text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})\) of \(\overline {\mathbb{Q}}\) on Belyi pairs \((X, \beta)\) induces a faithful action of \(\mathbb{G}\) on maps or on hypermaps, thus giving an alternative insight into this important and complicated group. We explain how these topics are closely related to certain congruence subgroups of the modular group and to elliptic curves.

Reviewer: G.Jones (Southampton)

##### MSC:

14H55 | Riemann surfaces; Weierstrass points; gap sequences |

11R32 | Galois theory |

05C10 | Planar graphs; geometric and topological aspects of graph theory |

11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |

14H30 | Coverings of curves, fundamental group |

11F06 | Structure of modular groups and generalizations; arithmetic groups |

20F65 | Geometric group theory |

30F10 | Compact Riemann surfaces and uniformization |