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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
##### Keywords:
Galois cover; inertia conjecture; wild ramification
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##### References:
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