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Polynomials with \(\text{PSL}(2)\) monodromy. (English) Zbl 1214.12003

Let \({\mathcal C}\) and \({\mathcal D}\) be smooth, projective, geometrically irreducible curves over a field \(k\) of characteristic \(p\geq 0\), and let \(f: {\mathcal C}\to{\mathcal D}\) be a separable morphism over \(k\) of degree \(d\geq 2\). Let \(k({\mathcal C})/k({\mathcal D})\) be the separable field extension corresponding to \(f\), and let \(E\) denotes its Galois closure. The arithmetic monodromy group of \(f\) is the group \(\text{Gal}(E/k({\mathcal D}))\). Letting \(l\) denote the algebraic closure of \(k\) in \(E\), the geometric monodromy group of \(f\) is \(\text{Gal}(E/l({\mathcal D}))\). A fundamental problem is to determine the possibilities for the monodromy groups and the ramification of such maps \(f\), where \({\mathcal D}\) is fixed and \({\mathcal C}\) or \(f\) varies.
In this paper, the authors determine all maps \(f\) having certain monodromy groups, subject to a constraint on the ramification. More explicitly, let \(q\) be a power of \(p\) and let \(u\) be transcendental over \(k\). The authors determine all polynomials \(f(X)\in k[X]\setminus k[X^p]\) of degree \(q(q- 1)/2\) for which the Galois group of \(f(X)- u\) over \(k(u)\) has a transitive normal subgroup isomorphic to \(\text{PSL}_2(q)\), subject to a certain ramification hypothesis. As a consequence, they describe all polynomials \(f\in k[X]\) such that the degree of \(f\) is not a power of \(p\) and \(f\) is functionally indecomposable over \(k\) but decomposes over an extension of \(k\). Moreover, except for one ramification configuration, they describe all indecomposable \(f\in k[X]\) such that \(\deg(f)\) is not a power of \(p\) and \(f\) is exceptional in the sense of the previous review [Ann. Math. (2) 172, No. 2, 1361–1390 (2010; Zbl 1214.12003)].

MSC:

12F20 Transcendental field extensions
12E05 Polynomials in general fields (irreducibility, etc.)

Citations:

Zbl 1214.12003
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