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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.
MSC:
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
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