The ‘obvious’ part of Belyi’s theorem and Riemann surfaces with many automorphisms.

*(English)*Zbl 0915.14021
Schneps, Leila (ed.) et al., Geometric Galois actions. 1. Around Grothendieck’s “Esquisse d’un programme”. Proceedings of the conference on geometry and arithmetic of moduli spaces, Luminy, France, August 1995. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 242, 97-112 (1997).

Many articles on Grothendieck dessins, Belyi functions, hypermaps on Riemann surfaces etc. have at least in part the character of survey articles or give new access to old material. The present article again belongs to this category of papers and tries to shed new light on some old subjects and to make their connection visible. The first subject is the well-known result of G. V. Belyi [Math. USSR, Izv. 14, 247-256 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 267-276 (1979; Zbl 0409.12012)]:

Theorem 1. A compact Riemann surface \(X\) is isomorphic to the Riemann surface \(C(\mathbb{C})\) consisting of the complex points of a nonsingular projective algebraic curve \(C\) defined over a number field if and only if there is a nonconstant meromorphic function \(\beta\) on \(X\) ramified over at most three points.

Such functions will be called Belyi functions. Of course we may assume that they are ramified over the three points 0, 1 and \(\infty\). The surprisingly simple algorithm found by Belyi to prove the ‘only if’ part of the theorem is reproduced in many later papers and will not be discussed here, we will care about the ‘if’ part only. For this proof Belyi refers to Weil’s criteria [A. Weil, Am. J. Math. 78, 509-524 (1956; Zbl 0072.16001)]. As a variant finally coming from the same source one may use the explicit knowledge of the Galois groups of the maximal extensions of \(\mathbb{C}(x)\) and \(\overline \mathbb{Q}(x)\) unramified outside three points [B. H. Matzat, “Konstruktive Galoistheorie”, Lect. Notes Math. 1284 (1987; Zbl 0634.12011)]. Both proofs rely on a heavy machinery and it is far from being obvious how to fit the problem precisely to the hypotheses of Weil’s criteria, and whether or not such a powerful tool is needed. I hope therefore that another version of this proof will be of some use, but the reader will recognize that we do not leave the neighbourhood of Weil’s ideas. In this part of the proof – lemmas 3 and 4 – old-fashioned and elementary but still vital and useful algebraic geometry is needed: Zariski topology, generic points and a specialization argument. Similar ideas often have been used in the literature: Shimura’s and Taniyama’s proof that abelian varieties with complex multiplication may be defined over a number field [G. Shimura and Y. Taniyama, “Complex multiplication of abelian varieties and its applications to number theory” (1961; Zbl 0112.03501); proposition 26] is an early example. In the present paper, certain coverings of \(X\) are the main topological tools. This part (up to lemma 2) is based on the rigidity of triangle groups.

As a last subject, the methods explained in this note give a new proof for the essential part of Popp’s result on a conjecture of Rauch about Riemann surfaces with many automorphisms (theorem 5) showing again the close connection of Belyi’s theorem to moduli problems. These Riemann surfaces with many automorphisms turn out to be of particular interest for Galois actions (theorem 7).

For the entire collection see [Zbl 0868.00041].

Theorem 1. A compact Riemann surface \(X\) is isomorphic to the Riemann surface \(C(\mathbb{C})\) consisting of the complex points of a nonsingular projective algebraic curve \(C\) defined over a number field if and only if there is a nonconstant meromorphic function \(\beta\) on \(X\) ramified over at most three points.

Such functions will be called Belyi functions. Of course we may assume that they are ramified over the three points 0, 1 and \(\infty\). The surprisingly simple algorithm found by Belyi to prove the ‘only if’ part of the theorem is reproduced in many later papers and will not be discussed here, we will care about the ‘if’ part only. For this proof Belyi refers to Weil’s criteria [A. Weil, Am. J. Math. 78, 509-524 (1956; Zbl 0072.16001)]. As a variant finally coming from the same source one may use the explicit knowledge of the Galois groups of the maximal extensions of \(\mathbb{C}(x)\) and \(\overline \mathbb{Q}(x)\) unramified outside three points [B. H. Matzat, “Konstruktive Galoistheorie”, Lect. Notes Math. 1284 (1987; Zbl 0634.12011)]. Both proofs rely on a heavy machinery and it is far from being obvious how to fit the problem precisely to the hypotheses of Weil’s criteria, and whether or not such a powerful tool is needed. I hope therefore that another version of this proof will be of some use, but the reader will recognize that we do not leave the neighbourhood of Weil’s ideas. In this part of the proof – lemmas 3 and 4 – old-fashioned and elementary but still vital and useful algebraic geometry is needed: Zariski topology, generic points and a specialization argument. Similar ideas often have been used in the literature: Shimura’s and Taniyama’s proof that abelian varieties with complex multiplication may be defined over a number field [G. Shimura and Y. Taniyama, “Complex multiplication of abelian varieties and its applications to number theory” (1961; Zbl 0112.03501); proposition 26] is an early example. In the present paper, certain coverings of \(X\) are the main topological tools. This part (up to lemma 2) is based on the rigidity of triangle groups.

As a last subject, the methods explained in this note give a new proof for the essential part of Popp’s result on a conjecture of Rauch about Riemann surfaces with many automorphisms (theorem 5) showing again the close connection of Belyi’s theorem to moduli problems. These Riemann surfaces with many automorphisms turn out to be of particular interest for Galois actions (theorem 7).

For the entire collection see [Zbl 0868.00041].

##### MSC:

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

30F10 | Compact Riemann surfaces and uniformization |

14H30 | Coverings of curves, fundamental group |