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Three point covers with bad reduction. (English) Zbl 1062.14038

Let \(X\) be a smooth projective curve over \(\mathbb{C}\). A celebrated theorem of Belyi states that \(X\) can be defined over a number field \(K\) if and only if there exists a finite map \(f: X \rightarrow \mathbb{P}^{1}\) ramified only above the three points \(\{0,1,\infty\}\). In such way, for every element \(\sigma\in\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) of the absolute Galois group of \(\mathbb{Q}\) we obtain a conjugate three point cover \(f^\sigma:X^\sigma\to\mathbb{P}^{1}\), which may or may not be isomorphic to \(f\). This yields a continuous action of \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on the set of isomorphism classes of three point covers. Hence we can associate to \(f\) the number field \(K\) such that \(\text{Gal}(\overline{\mathbb{Q}}/K)\) is precisely the stabilizer of the isomorphism class of \(f\). The field \(K\) is called the field of moduli of \(f\). Very little is known about the correspondence between three point covers and their associated field of moduli. One of the most general result that we known about \(K\) is a theorem of of S. Beckmann [J. Algebra 125, 236–255 (1989; Zbl 0698.14024)] which says that the extension \(K/{\mathbb{Q}}\) is unramified outside the set of primes dividing the order of the group \(G\). This result is related to the fact that \(f\) has good reduction at each prime ideal of \(K\) dividing \(p\). In the case that \(f\) has bad reduction, the author of the paper under review obtains the following
Theorem. Let \(f: X \rightarrow \mathbb{P}^{1}\) be a three point cover, with field of moduli \(K\) and monodromy group \(G\). Let \(p\) be a prime number which strictly divides the order of \(G\), i.e. \(p^{2}\) does not divide the order of \(G\). Then \(p\) is at most tamely ramified in the extension \(K/{\mathbb{Q}}\).
This theorem follows easily from theorem 2 and theorem 3 of the same paper. Theorem 2 has an involved statement and concerns a stable reduction \(\overline{f}: \overline{X} \rightarrow \overline{Y}\) of a three point cover \(f: X \rightarrow \mathbb{P}^{1}\). Theorem 3 concerns a stable reduction \(\overline{f}: \overline{X} \rightarrow \overline{Y}\) which is a special \(G\)-map over \(k\). To give a general idea, it suffice to say that a special \(G\)-map is a finite map \(\overline{f}: \overline{X} \rightarrow \overline{Y}\) between stable curves, together with an embedding of \(G \) in \(\text{Aut}(\overline{X}/\overline{Y})\), which admits a compatible special deformation datum \((\overline{Z}_{0},\omega_{0})\). Then theorem 3 says that if \(\overline{f}: \overline{Y} \rightarrow \overline{X}\) is a special \(G\)-map defined over an algebraically closed field \(k\) of characteristic \(p>0\), there exists a three point Galois cover \(f_{K}: Y_{K} \rightarrow X_{K}\) whose stable reduction is isomorphic to \(\overline{f}: \overline{Y} \rightarrow \overline{X}\), and every lift of \(\overline{f}\) can be defined over a finite extension \(K/K_{0}\) which is at most tamely ramified.

MSC:

14H30 Coverings of curves, fundamental group
11G20 Curves over finite and local fields

Citations:

Zbl 0698.14024
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References:

[1] Sybilla Beckmann, Ramified primes in the field of moduli of branched coverings of curves, J. Algebra 125 (1989), no. 1, 236 – 255. · Zbl 0698.14024 · doi:10.1016/0021-8693(89)90303-7
[2] José Bertin and Ariane Mézard, Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques, Invent. Math. 141 (2000), no. 1, 195 – 238 (French, with English summary). · Zbl 0993.14014 · doi:10.1007/s002220000071
[3] I.I. Bouw and R.J Pries, Rigidity, reduction, and ramification, To appear in Math. Ann. · Zbl 1029.14010
[4] I.I. Bouw and S. Wewers, Reduction of covers and Hurwitz spaces, arXiv:math. AG/0005120, 2000. · Zbl 1058.14050
[5] I.I. Bouw and S. Wewers, Stable reduction of modular curves, arXiv:math.AG/0210363, 2002. · Zbl 1147.11316
[6] Robert Coleman and William McCallum, Stable reduction of Fermat curves and Jacobi sum Hecke characters, J. Reine Angew. Math. 385 (1988), 41 – 101. · Zbl 0654.12003
[7] Pierre Dèbes and Jean-Claude Douai, Algebraic covers: field of moduli versus field of definition, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 3, 303 – 338 (English, with English and French summaries). · Zbl 0906.12001 · doi:10.1016/S0012-9593(97)89922-3
[8] Daniel Ferrand and Michel Raynaud, Fibres formelles d’un anneau local noethérien, Ann. Sci. École Norm. Sup. (4) 3 (1970), 295 – 311 (French). · Zbl 0204.36601
[9] Barry Green and Michel Matignon, Order \? automorphisms of the open disc of a \?-adic field, J. Amer. Math. Soc. 12 (1999), no. 1, 269 – 303. · Zbl 0923.14007
[10] Y. Henrio, Arbres de Hurwitz et automorphismes d’ordre \(p\) des disques et des couronnes \(p\)-adic formels, To appear in: Comp. Math., available at arXiv:math.AG/0011098, 2000.
[11] Finn F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks \?_{\?,\?}, Math. Scand. 52 (1983), no. 2, 161 – 199. , https://doi.org/10.7146/math.scand.a-12001 Finn F. Knudsen, The projectivity of the moduli space of stable curves. III. The line bundles on \?_{\?,\?}, and a proof of the projectivity of \overline\?_{\?,\?} in characteristic 0, Math. Scand. 52 (1983), no. 2, 200 – 212. · Zbl 0544.14021 · doi:10.7146/math.scand.a-12002
[12] B. Koeck, Belyi’s theorem revisited, arXiv:math.AG/0108222.
[13] Claus Lehr, Reduction of \?-cyclic covers of the projective line, Manuscripta Math. 106 (2001), no. 2, 151 – 175. · Zbl 1065.14508 · doi:10.1007/s002290100190
[14] Gunter Malle, Fields of definition of some three point ramified field extensions, The Grothendieck theory of dessins d’enfants (Luminy, 1993) London Math. Soc. Lecture Note Ser., vol. 200, Cambridge Univ. Press, Cambridge, 1994, pp. 147 – 168. · Zbl 0871.14021
[15] James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. · Zbl 0433.14012
[16] Rachel J. Pries, Construction of covers with formal and rigid geometry, Courbes semi-stables et groupe fondamental en géométrie algébrique (Luminy, 1998) Progr. Math., vol. 187, Birkhäuser, Basel, 2000, pp. 157 – 167. · Zbl 0978.14018
[17] R.J. Pries, Families of wildly ramified covers of curves, Amer. J. Math. 124 (2002), no. 4, 737-768. · Zbl 1059.14033
[18] Michel Raynaud, Schémas en groupes de type (\?,…,\?), Bull. Soc. Math. France 102 (1974), 241 – 280 (French). · Zbl 0325.14020
[19] Michel Raynaud, \?-groupes et réduction semi-stable des courbes, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 179 – 197 (French). · Zbl 0722.14013 · doi:10.1007/978-0-8176-4576-2_7
[20] Michel Raynaud, Spécialisation des revêtements en caractéristique \?>0, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 1, 87 – 126 (French, with English and French summaries). · Zbl 0999.14004 · doi:10.1016/S0012-9593(99)80010-X
[21] M. Saïdi, Torsors under finite and flat group schemes of rank \(p\)with Galois action, arXiv:math.AG/0106246, 2001.
[22] Jean-Pierre Serre, Corps locaux, Hermann, Paris, 1968 (French). Deuxième édition; Publications de l’Université de Nancago, No. VIII. · Zbl 1095.11504
[23] S. Wewers, Formal deformations of curves with group scheme action, arXiv:math.AG/0212145, 2002.
[24] S. Wewers, Reduction and lifting of special metacyclic covers, Ann. Scient. Éc. Norm. Sup. 36 (2003), 113-138. · Zbl 1042.14005
[25] Leonardo Zapponi, The arithmetic of prime degree trees, Int. Math. Res. Not. 4 (2002), 211 – 219. · Zbl 1074.14521 · doi:10.1155/S1073792802011054
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