31 research outputs found
Good reduction of three-point Galois covers
Michel Raynaud gave a criterion for a three-point G-cover f : Y \rightarrow X
= P^1, defined over a p-adic field K, to have good reduction. In particular, if
the order of a p-Sylow subgroup of G is p, and the number of conjugacy classes
of elements of order p is greater than the absolute ramification index e of K,
then f has potentially good reduction. We give a different proof of this
criterion, which extends to the case where G has an arbitrarily large cyclic
p-Sylow subgroup, answering a question of Raynaud. We then use the criterion to
give a family of examples of three-point covers with good reduction to
characteristic p and arbitrarily large p-Sylow subgroups.Comment: Minor revisions, to appear in Algebraic Geometry. 16 page
On Colmez's product formula for periods of CM-abelian varieties
Colmez conjectured a product formula for periods of abelian varieties with
complex multiplication by a field K, analogous to the standard product formula
in algebraic number theory. He proved this conjecture up to a rational power of
2 for K/Q abelian. In this paper, we complete the proof of Colmez for K/Q
abelian by eliminating this power of 2. Our proof relies on analyzing the
Galois action on the De Rham cohomology of Fermat curves in mixed
characteristic (0, 2), which in turn relies on understanding the stable
reduction of Z/2^n-covers of the projective line, branched at three points.Comment: Final version, 16 pp., to appear in Math. Ann. The final publication
is available at http://www.springerlink.co
Fields of moduli of three-point G-covers with cyclic p-Sylow, I
We examine in detail the stable reduction of Galois covers of the projective
line over a complete discrete valuation field of mixed characteristic (0, p),
where G has a cyclic p-Sylow subgroup of order p^n. If G is further assumed to
be p-solvable (i.e., G has no nonabelian simple composition factors with order
divisible by p), we obtain the following consequence: Suppose f: Y --> P^1 is a
three-point G-Galois cover defined over the complex numbers. Then the nth
higher ramification groups above p for the upper numbering of the (Galois
closure of the) extension K/Q vanish, where K is the field of moduli of f. This
extends work of Beckmann and Wewers. Additionally, we completely describe the
stable model of a general three-point Z/p^n-cover, where p > 2.Comment: Major reorganization. In particular, the former Appendix C has been
spun off and is now arxiv:1109.4776. Now 42 page
Wild ramification kinks
Given a branched cover between smooth projective curves over a
non-archimedian mixed-characteristic local field and an open rigid disk
, we study the question under which conditions the inverse image
is again an open disk. More generally, if the cover varies in
an analytic family, is this true at least for some member of the family? Our
main result gives a criterion for this to happen.Comment: Final version, to appear in Research in the Mathematical Sciences. 29
page