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Killing wild ramification. (English) Zbl 1306.14013
Let \(p\) be prime and let \(G\) be a quasi-\(p\) group. Thus \(G\) is a finite group which is generated by its elements of \(p\)-power order. Let \(I\) be a subgroup of \(G\) whose conjugates generate \(G\) such that \(I\) contains a normal Sylow \(p\)-subgroup \(H\) with cyclic quotient \(I/H\). Let \(k\) be an algebraically closed field of characteristic \(p\). Abhyankar’s affine inertia conjecture [S. S. Abhyankar, Bull. Am. Math. Soc., New Ser. 38, No. 2, 131–169 (2001; Zbl 0999.12003), p. 156] asserts that there is a \(G\)-Galois cover \(X\rightarrow{\mathbb P}^1\) of irreducible smooth curves over \(k\), branched only at \(\infty\), which has \(I\) as an inertia group over \(\infty\). Let \(X\rightarrow{\mathbb P}^1\) be a \(G\)-Galois cover, branched only at \(\infty\), such that a \(p\)-subgroup \(P\) of \(G\) occurs as an inertia group over \(\infty\). D. Harbater [Am. J. Math. 115, No. 3, 487–508 (1993; Zbl 0790.14027), Theorem 2] showed that if \(Q\) is a \(p\)-subgroup of \(G\) containing \(P\), then there is a \(G\)-Galois cover \(Y\rightarrow{\mathbb P}^1\), branched only at \(\infty\), which has \(Q\) as an inertia group over \(\infty\). Suppose that \(G\) does not map onto any nontrivial quotient of \(P\). The present paper shows that in this case there is a \(G\)-Galois cover \(Z\rightarrow{\mathbb P}^1\), branched only at \(\infty\), which has the lower ramification subgroup \(P_2\) of \(P\) as an inertia group over \(\infty\). Hence if \(P_2\) is a proper subgroup of \(P\) this allows one to replace the inertia group \(P\) with a smaller subgroup of \(G\), rather than a larger one. This result can be viewed as extending Abhyankar’s lemma to a setting involving wild ramification.

MSC:
14H30 Coverings of curves, fundamental group
11S15 Ramification and extension theory
11G20 Curves over finite and local fields
11R32 Galois theory
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