Stefan Wewers
Special covers are metacyclic covers of the projective line, with Galois group of order pm, which have a specific type of bad reduction to characteristic p. Such covers arise in the study of the arithmetic of Galois covers of the projective line with three branch points. Our results provide a classification of all special covers in terms of certain lifting data in characteristic p.
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We give a necessary and sufficient condition for a modular representation of a group $G=C_{p^h} \rtimes C_m$ in a field of characteristic zero to be lifted to a representation over local principal ideal domain of characteristic zero containing the $p^h$ roots of unity.
We compute the stable reduction of some Galois covers of the projective line branched at three points. These covers are constructed using Hurwitz spaces parameterizing metacyclic covers. The reduction is determined by a hypergeometric differential equation. This generalizes the result of Deligne- Rapoport on the reduction of the modular curve X(p).
Let $f:X\to Y$ be a finite ramified Galois covering of algebraic varieties defined over the complex numbers. In this paper, we prove some structure theorems for such coverings in the case that the non-abelian Galois group of the cover is metabelian (in particular metacyclic). Our results extend the previous ones, obtained by several authors, in the case of abelian and dihedral Galois coverings. In analogy with the abelian (and generalizing the dihedral) case, we find "building data" of metabelian covers from which the cover can be reconstructed. In addition to analyzing several examples, we co…
We construct and study two series of curves whose Jacobians admit complex multiplication. The curves arise as quotients of Galois coverings of the projective line with Galois group metacyclic groups $G_{q,3}$ of order $3q$ with $q \equiv 1 \mod 3$ an odd prime, and $G_m$ of order $2^{m+1}$. The complex multiplications arise as quotients of double coset algebras of the Galois groups of these coverings. We work out the CM-types and show that the Jacobians are simple abelian varieties.
We study the components of the Hurwitz scheme of ramified coverings of $\mathbb{P}^1$ with monodromy given by the alternating group $A_6$ and elements in the conjugacy class of product of two disjoint cycles. In order to detect the connected components of the Hurwitz scheme, inspired by the case of the spin structures studied by Fried for the $3$-cycles, we use as invariant the lifting to the Valentiner group, triple covering of $A_6$. We prove that the Hurwitz scheme has two irreducible components when the genus of the covering is greater than zero, in accordance with the asymptotic solution …
This work continues the study of residually wild morphisms $f\colon Y\to X$ of Berkovich curves initiated by Cohen, Temkin and Trushin in [CTT16]. The different function $δ_f$ introduced in [CTT16] is the primary discrete invariant of such covers. When $f$ is not residually tame, it provides a non-trivial enhancement of the classical invariant of $f$ consisting of morphisms of reductions $\widetilde{f}\colon \widetilde{Y}\to\widetilde{X}$ and metric skeletons $Γ_f\colon Γ_Y\toΓ_X$. In this paper we interpret $δ_f$ as the norm of the canonical trace section $τ_f$ of the dualizing sheaf $ω_f$, a…
In this paper we investigate the problem of lifting of Galois covers between algebraic curves from characteristic p>0 to characteristic 0. We prove a refined version of the main result of Garuti concerning this problem in [Ga]. We formulate a refined version of the Oort conjecture on liftings of cyclic Galois covers between curves. We introduce the notion of fake liftings of cyclic Galois covers between curves, their existence would contradict the Oort conjecture, and we study the geometry of their semi-stable models. Finally, we introduce and investigate on some examples the smoothening proce…
Let $k$ be an algebraically closed field of characteristic $p > 0$. We consider the problem of lifting $p$-cyclic covers of $\Proj_k$ as $p$-cyclic covers of the projective line over some DVR under the condition that the wild monodromy is maximal. We answer positively the question for covers birational to $w^ p-w=tR(t)$ for some additive polynomial $R(t)$.
We construct the moduli space of Enriques surfaces in positive characteristic and eventually over the integers, and determine its local and global structure. As an application, we show lifting of Enriques surfaces to characteristic zero. The key observation is that the canonical double cover of an Enriques surface is birational to the complete intersection of three quadrics in IP^5, even in characteristic 2.
Lifting Elementary Abelian Covers of Curves
本文针对特征p域上曲线的初等阿贝尔p-群伽罗瓦覆盖的提升问题，给出依赖p-循环子覆盖提升分支轨迹的组合判据，研究分支点在特殊纤维的合并规律，并分析多类(ℤ/2)³-覆盖的提升情形。
We prove a lifting theorem for odd Frattini covers of finite groups. Using this, we characterize solvable groups and more generally p-solvable groups in terms of containing a triple of elements of distinct prime power orders with product 1. This is also related to various questions about Fried's modular tower program and properties of Hurwitz spaces of covers of curves.
Lifting Prym Divisors on a Bielliptic Loop of Loops
本文针对双椭圆曲线的平展二重覆盖，证明其环之环上的普赖姆除子可提升为原曲线的同秩普赖姆除子，进而确定普赖姆-布里尔-诺特簇的维数、既约不可约性及上同调类。
The refined local lifting problem for cyclic covers of order four
本文研究了特征2代数闭域上四次循环覆盖的精细局部提升问题，证明了在给定子覆盖提升的条件下存在有限扩张及相应的整体提升，验证了Saïdi的精细提升猜想并揭示了野分歧Galois覆盖模空间的性质。
We solve the local lifting problem for the alternating group A_4, thus showing that it is a local Oort group. Specifically, if k is an algebraically closed field of characteristic 2, we prove that every A_4-extension of k[[s]] lifts to characteristic zero. As a consequence, every A_4-branched cover of smooth projective curves in characteristic 2 lifts to characteristic zero.
For a smooth finite cyclic covering over a projective space of dimension greater than one, we show that the group of automorphisms acts faithfully on the cohomology except for a few cases. In characteristic zero, we study the equivariant deformation theory and automorphism groups for complex cyclic coverings. The proof uses the decomposition of the sheaf of differential forms due to Esnault and Viehweg. In positive characteristics, a lifting criterion of automorphisms reduces the faithfulness problem to characteristic zero. To apply this criterion, we prove the degeneration of the Hodge-de Rha…
We study the Oort groups for a prime p, i.e. finite groups G such that every G-Galois branched cover of smooth curves over an algebraically closed field of characteristic p lifts to a G-cover of curves in characteristic 0. We prove that all Oort groups lie in a particular class of finite groups that we characterize, with equality of classes under a conjecture about local liftings. We prove this equality unconditionally if the order of G is not divisible by 2p^2. We also treat the local lifting problem and relate it to the global problem.
We give a group-theoretic description of the parity of a pull-back of a theta characteristic under a branched covering. It involves lifting monodromy of the covering to the semidirect product of the symmetric and Clifford groups, known as the Sergeev group. As an application, we enumerate torus coverings with respect to their ramification and parity and, in particular, show that the corresponding all-degree generating functions are quasimodular forms.
A necessary condition for liftings of positive characteristic varieties with finite fundamental groups
研究具有有限基本群的正特征簇提升到特征零的必要条件，基于平展同伦理论和沃尔有限性障碍，通过定义模l有限支配性概念，证明簇提升的链复形条件，还扩展了沃尔有限性障碍讨论。
We construct a 4-dimensional family of surfaces of general type with p_g=0 and K^2=3 and fundamental group Z/2xQ_8, where Q_8 is the quaternion group. The family constructed contains the Burniat surfaces with K^2=3. Additionally, we construct the universal coverings of the surfaces in our family as complete intersections on (\PP^1)^4 and we also give an action of Z/2xQ_8 on (\PP^1)^4 lifting the natural action on the surfaces. The strategy is the following. We consider an étale (Z/2)^3-cover T of a surface with p_g=0 and K^2=3 and assume that it may be embedded in a Fano 3-fold V. We construct…
The paper offers versions of Hilbert's Irreducibility Theorem for the lifting of points in a cyclic subgroup of an algebraic group to a ramified cover. A version of Bertini Theorem in this context is also obtained.
Contents: Rational functions with given monodromy on generic curves (I. Bouw & S. Wewers); Can deformation rings of group representations not be local complete intersections? (T. Chinburg); Lifting an automorphism group to finite characteristic (G. Cornelissen); Flat connections and representations of the fundamental group in characteristic p>0 (C. Gasbarri) Questions on p-torsion of hyperelliptic curves (D. Glass & R. Pries); Automorphisms of curves and stable reduction (C. Lehr & M. Matignon); p-groups as automorphism groups of curves in characteristic p; Lifting Galois covers of smooth curv…
Kummer-Artin-Schreier-Witt Theory
本文提出通过库默-阿廷-施雷尔-维特理论，将正特征中的阿廷-施雷尔-维特同态显式提升到零特征，并探讨其在不分离扩张和斯万展开式研究中的应用。
We prove the existence of noncrossed product and indecomposable division algebras over the function field of a smooth p-adic curve, especially when the curve does not admit a smooth model over Z_p. Thus we generalize arXiv 0907.0670. To make our constructions, we investigate the lifting of cohomology classes from the total fraction ring of the closed fiber to the function field of the curve, over an arbitrary discrete valuation ring of mixed characteristic.
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