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.


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


Zbl 0698.14024
Full Text: DOI arXiv


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