Conductors of wildly ramified covers. III.
Let \(X\) be a smooth connected proper curve with marked points \(\{x_i\}\) over an algebraically closed field \(k\) of characteristic \(p\). Consider a Galois cover \(\phi : Y \to X\) of smooth connected curves branched only at \(\{x_i\}\). Abhyankar's conjecture (proved by \textit{M. Raynaud}, [Invent. Math. 116, 425--462 (1994; Zbl 0798.14013)] and \textit{D. Harbater} [Invent. Math. 117, 1--25 (1994; Zbl 0805.14014)]) determines exactly which groups \(G\) can be the Galois group of \(\phi\). An open problem is to determine which inertia groups and filtrations of higher ramification groups can be realized for such a cover \(\phi\). In this article, the author deals with the case \(\phi : Y \to \mathbb{P}^1\) is branched at only one point. Such covers exist if and only if \(G\) is a quasi-\(p\) group, which means that \(G\) is generated by \(p\)-groups. \textit{D. Harbater} [Am. J. Math. 115, 487--508 (1993; Zbl 0790.14027)] proved that the Sylow \(p\)-subgroups of \(G\) can be realized as the inertia groups of such a cover \(\phi\). Under the assumption that the Sylow \(p\)-subgroups of \(G\) have order \(p\), the filtration of higher ramification groups is determined by one integer \(j\) for which \(p \nmid j\) namely by the lower jump or conductor. The author shows that all sufficiently large conductors occur and for \(p \neq 2\) gives an explicit bound for which conductors are sufficiently large. Theorem. Let \(G\) be a finite quasi-\(p\) group whose Sylow \(p\)-subgroups have order \(p \neq 2\). There exists an integer \(J\) depending explicitly on \(p\), the \(p\)-weight of \(G\), and the exponent of the normalizer of a Sylow \(p\)-subgroup of \(G\) with the following property: if \(j \geq J\) and \(p \nmid j\) then there exists a \(G\)-Galois cover \(\phi : Y \to \mathbb{P}^1\) branched at only one point over which it has inertia group \(\mathbb{Z}/p\) and conductor \(j\). The \(p\)-weight of \(G\) is defined in the following way. Let \(S\) be a chosen Sylow \(p\)-subgroup of \(G\). Let \(G(S)\) be the subgroup generated by all proper quasi-\(p\) subgroups \(G'\) such that \(G' \cap S\) is a Sylow \(p\)-subgroup of \(G'\). The group \(G\) is \(p\)-pure if \(G(S) \neq G\). Now consider all subgroups \(G' \subset G\) such that \(G'\) is quasi-\(p\) and \(p\)-pure and such that \(G' \cap S\) is a Sylow \(p\)-subgroup of \(G'\). The \(p\)-weight \(\omega(G)\) of \(G\) is the minimal number of such subgroups \(G'\) of \(G\) which are needed to generate \(G\) (it is independent of the choice of \(S\)). The integer \(J\) is then given by the following definition: let \(m_e\) be the exponent of the normalizer \(N_G(S)\) of \(S\) in \(G\) divided by \(p\). One has \(J=m_e(2+1/(p-1))\omega(G)\) if \(p \nmid \omega(G)\) and \(J=m_e(2+1/(p-1)) \omega(G)+2\) otherwise.
scientific article; zbMATH DE number 2101105