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On the inertia conjecture for alternating group covers. (English) Zbl 1442.14097
Summary: Let \(G_1\) and \(G_2\) be perfect groups such that there exist connected \(G_1\)-Galois and \(G_2\)-Galois étale covers of the affine line over an algebraically closed field of characteristic \(p > 0\) with the cyclic \(p\)-groups \(P_1\) and \(P_2\) as the inertia groups above \(\infty\), respectively. Then we show that there is a connected \(G_1 \times G_2\)-Galois étale cover of the affine line with an inertia group \(I\) above \(\infty\) where \(I\) is a cyclic subgroup of \(P_1 \times P_2\) of index \(p\). As a consequence, it is shown that the wild part of the Inertia Conjecture is true for any product of Alternating groups, each of degree \(p\) or coprime to \(p\). For \(d\) a multiple of \(p\), a new étale \(A_d\)-cover of the affine line is obtained using an explicit equation, and it is shown that this cover has the minimal possible upper jump.
14H30 Coverings of curves, fundamental group
13B05 Galois theory and commutative ring extensions
14G17 Positive characteristic ground fields in algebraic geometry
11S15 Ramification and extension theory
Full Text: DOI
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