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Noncongruence subgroups, covers and drawings. (English) Zbl 0930.11024

Schneps, Leila (ed.), The Grothendieck theory of dessins d’enfants. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 200, 25-46 (1994).
From the paper: The theory of congruence subgroups, more precisely the theory of the action of congruence subgroups of the modular group on the upper half plane, is an area of mathematics in which the mathematical structure is well understood and wonderfully intricate. In contrast, the arithmetic theory of subgroups which are not congruence subgroups is surprisingly little developed. In a neighbouring area, there is a beautiful corpus of recent work concerned with the arithmetic of Galois coverings of the projective line, ramified in a prescribed way above a finite set of places; a motivation for this has been its application to the inverse Galois problem. This essay is a summary of various talks given by the author on the subject during the past couple of years.
Theorem 1. For each positive integer \(n\), the following families of objects are in 1-1 correspondence:
(i) Triples \(({\mathcal R},\varphi,O)\) where \({\mathcal R}\) is an \(n\)-sheeted Riemann surface, \(\varphi:{\mathcal R}\to\overline\mathbb{C}\) is a covering map branched at most above \(\{\infty,0,k\}\), and \(O\) is a point of \({\mathcal R}\) above 0;
(ii) Quadruples \((\sigma_\infty,\sigma_0,\sigma_1;\nu)\) where \(\sigma_\infty,\sigma_0,\sigma_1\) are permutations of \(S_n\) such that \(\sigma_\infty\sigma_0\sigma_1=\text{id}\) and such that the group generated by \(\sigma_0,\sigma_1\) is transitive on the symbols permuted by \(S_n\), and \(\nu\) is a marked cycle of \(\sigma_\infty\); all modulo equivalence corresponding to simultaneous conjugation by an element of \(S_n\);
(iii) Subgroups \(\Gamma\leq\Gamma(2)\) of index \(n\), modulo conjugacy by translation;
(iv) Drawings with \(n\) edges.
Here a drawing is a finite connected 1-complex which is the skeleton of an oriented 2-complex. It has two sets of vertices, coloured respectively black and white, and a set of edges. Each edge is incident to a single vertex of each colour, and for each vertex there is a cyclic ordering of the edges incident to it. These drawings are of course pretty much the same as what G. E. Shabat and V. A. Voevodsky [The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 199-227 (1990; Zbl 0790.14026)] call a dessin: their dessins are a particular case of drawings. Conversely, every drawing may be made into a dessin by re-colouring all the vertices black, and putting a new white vertex in the middle of each edge.
Having set up his apparatus, the author lists three questions, essentially the ones asked by A. O. L. Atkin and H. P. F. Swinnerton-Dyer [in: Combinatorics, Proc. Symp. Pure Math. 19, 1-25 (1971; Zbl 0235.10015)].
Question 0. Describe how to list the subgroups of \(\Gamma(2)\), in order (say) of increasing index.
Question 1. The Riemann surfaces \({\mathcal R}\) are algebraic curves, and the projections \(\varphi\) are algebraic maps. What can we say about the equations of these curves and maps; in particular, what is their field of definition?
Question 2. For any subgroup \(\Gamma\) of finite index in \(\Gamma(1)\), and any integer \(k\geq 1\), we may define cusp forms of weight \(2k\) on \(\mathbb{H}/\Gamma\). Such a form \(f(z)\) has a Fourier expansion \(f(z)=\sum^\infty_1a_rq^r_\mu\), where \(q_\mu=\exp(2\pi iz/\mu)\) and \(z\to z+\mu\) is the least translation in \(\Gamma\). What can be said about the coefficients \(a_n\)? (In the congruence subgroup case, the Hecke algebra action implies that there is a basis of ‘eigenforms’, whose coefficients are multiplicative.)
Question 0 has been answered in effect by (ii) of the theorem 1 since it is not too hard to train a computer to list pairs of permutations up to conjugacy. In section 2, it is recalled what is known in general about the nature of the curves \({\mathcal R}\) and the covering maps \(\varphi\). Almost the only answers to question 2 are due to A. J. Scholl [cf. Invent. Math. 79, 49-77 (1985; Zbl 0542.10022) and J. Reine Angew. Math. 392, 1-15 (1988; Zbl 0647.10022)] recalling his results in section 3. Finally, in section 4, a little bit is said about calculation, and tables of examples are given.
For the entire collection see [Zbl 0798.00001].

MSC:

11F06 Structure of modular groups and generalizations; arithmetic groups
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
14H55 Riemann surfaces; Weierstrass points; gap sequences
14E20 Coverings in algebraic geometry
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