Three point covers with bad reduction

Abstract

We study Galois covers of the projective line branched at three points with bad reduction to characteristic p, under the condition that p strictly divides the order of the Galois group. As an application of our results, we prove that the field of moduli of such a cover is at most tamely ramified at p.

Reduction and lifting of special metacyclic covers

“…This theorem gives us a concrete way to construct special H-covers and, together with Theorem A, yields a complete classification of special covers (up to solving the equations satisfied by the points τ i ). Again, this result has nice applications to the arithmetic of three point covers, see [16].…”

“…Theorem A essentially says that the stable reduction of a three point Galois cover of the projective line is as simple as one can expect it to be. This result turns out to be very useful to study the arithmetic of such covers, see [16]. Let us mention at this point that the 'three point condition' is essential for Theorem A to hold.…”

“…The results of Section 2 and Section 3 give, in some sense, a complete classification of all special covers and a thorough understanding of their arithmetic. These results are applied in [16] to the study of the arithmetic of three point Galois covers of P 1 with bad reduction. In particular, upper and lower bounds on the ramification index of p in the field of moduli of such covers are given, sharpening the results of [10].…”

Arithmetic descent of specializations of Galois covers

“…(Especially important is the technique of rigidity, developed by Belyi, Feit, Fried, Shih, Thompson, Matzat and others; see [Völ96], [Ser08], and [MM99] for details. There has also been research aimed at understanding how the field of moduli depends on the group, as well as on topological data; see for example [Bec89], [Flo04], [Wew03], [Obu12], [Obu13], and [Has13].) In this paper we study an alternative method of attack.…”

Conductors of wild extensions of local fields, especially in mixed characteristic $(0,2)$

“…Let f : Y → P 1 be a G-Galois cover of P 1 branched at 0, 1, and ∞, a priori defined over the algebraic closure of K 0 . If a p-Sylow subgroup of G is of order p, then it turns out that f can in fact be defined over a tame extension of K 0 ( [Wew03b]). However, if a p-Sylow subgroup of G is cyclic of order p r , then the best that can be proven at the moment, especially when p is small, is that f can often be defined over a field of the form K c ( p c √ a)/K 0 , where a ∈ K c .…”