Rigidity, reduction, and ramification. (English) Zbl 1029.14010

Let \(k\) be an algebraically closed field of characteristic \(p>2\). Let \(G\) be a finite quasi-\(p\) group. The “inertia conjecture” asserts that if \(I\) is a subgroup of \(G\) which is an extension of a cyclic group of order prime-to-\(p\) by a \(p\)-group \(P\), whose conjugates generate \(G\), then \(I\) occurs as inertia group for some \(G\)-Galois cover \(f : Y \rightarrow \mathbb{P}^1_k\) branched only at \(\infty\). The authors prove that the inertia conjecture is true for the groups \(A_p\) and \(\text{PSL}_2(p)\), giving the first serious evidence for this conjecture. In addition, they prove that the set of conductors that can be realized depends on the group. The main technique is to study covers \(f\) in characteristic zero such that \(p\) divides the ramification indices. Such a cover has bad reduction to characteristic \(p\). Under some conditions on the Galois group of \(f\), one component of the stable reduction of \(f\) yields a \(G\)-Galois cover of \(P_k^1\) which is wildly ramified above only one point. For \(G=\text{PSL}_2(p)\), the authors use the reduction of the modular curve \(X(2p)\), and for \(G=A_p\), they use the existence of subgroups with index \(p\).


14H30 Coverings of curves, fundamental group
14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
Full Text: DOI