Universal abelian covers of certain surface singularities

Every normal complex surface singularity with -homology sphere link has a universal abelian cover. It has been conjectured by Neumann and Wahl that the universal abelian cover of a rational or minimally elliptic singularity is a complete intersection singularity defined by a system of ``splice diagram equations'. In this paper we introduce a Neumann-Wahl system, which is an analogue of the system of splice diagram equations, and prove the following. If (X, o) is a rational or minimally elliptic singularity, then its universal abelian cover (Y, o) is an equisingular deformation of an isolated complete intersection singularity (Y₀, o) defined by a Neumann-Wahl system. Furthermore, if G denotes the Galois group of the covering Y → X, then G also acts on Y₀ and X is an equisingular deformation of the quotient Y₀/G. Dedicated to Professor Jonathan Wahl on his sixtieth birthday. This research was partially supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

Universal deformation rings need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. To cite this article: F.M. Bleher, T. Chinburg, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Déformations formelles de revêtements: un principe local-global

LetC be a generically smooth, locally complete intersection curve defined over an algebraically closed fieldk of characteristicp≥0. LetG⊃ Aut _k C be a finite group of automorphisms ofC. We develop a theory ofG-equivariant deformations of the Galois coverC→C/G. We give a thorough study of the local obstructions, those localized at singular or widely ramified points, to deform equivariantly a cover. As an application, we discuss the case ofG-equivariant deformations of semistable curves.      

Universal deformation rings need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose (unrestricted) universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. Finally, we discuss bounds on the singularities of universal deformation rings of representations of finite groups in terms of the nilpotency of the associated defect groups. The first author was supported in part by NSF Grant DMS01-39737 and NSA Grant H98230-06-1-0021. The second author was supported in part by NSF Grants DMS00-70433 and DMS05-00106.     

The Valentiner group as Galois group

We obtain the complete set of solutions to the Galois embedding problem given by the Valentiner group as a triple cover of the alternating group .

     

A conjecture on arithmetic fundamental groups

The conjecture is the following: Over an algebraic variety over a finite field, the geometric monodromy group of every smooth is finite. We indicate how to prove this for rank 2, using results of Drinfeld. We also show that the conjecture implies that certain deformation rings of Galois representations are complete intersection rings. This material is based upon work supported by the National Science Foundation under Grant No. 9970049.     

Killing wild ramification

We compute the inertia group of the compositum of wildly ramified Galois covers. It is used to show that even the p-part of the inertia group of a Galois cover of ?¹ branched only at infinity can be reduced if there is a jump in the lower ramification filtration at two and a certain linear disjointness statement holds.     

Galois covers of the open p-adic disc

This paper investigates Galois branched covers of the open p-adic disc and their reductions to characteristic p. Using the field of norms functor of Fontaine and Wintenberger, we show that the special fiber of a Galois cover is determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. As applications, we derive a criterion for good reduction in the abelian case, and give an arithmetic reformulation of the local Oort Conjecture concerning the liftability of cyclic covers of germs of curves.     

Tertiary decompositions in lattice modules

The theory of associated prime ideals of anR-module, and of tertiary decompositions, generalizes toL-modules, whereL is a complete modular lattice and anL-moduleM is a complete modular lattice together with an appropriate module actionp:L×MM. Given appropriate chain conditions onL andM, the theory of associated prime ideals, existence and uniqueness properties for tertiary decompositions, and a form of the Krull intersection theorem all hold in generalized form. If more stringent conditions apply, the theory reduces to a generalized theory of primary decompositions and a second uniqueness theorem holds. The theory can be applied to congruence lattices of algebras in congruence-modular varieties of algebras, using the generalized commutator operation. An important special case is the theory of finite groups, where the descending chain condition allows a natural choice of a distinguished tertiary decomposition and this yields a canonical decomposition of any finite group as a subdirect product of cotertiary finite groups. The group-theoretic application of the tertiary theory yields elementary structure theorems about Galois extensions of fields, where the tertiary decomposition of the Galois group transforms into a representation of a Galois extension as a compositum. For example, given a fieldF, there are distinguished tertiary field extensions ofF, of which all other finite Galois extensions ofF are compositums.Presented by Bjarni Jónsson.     

Fuzzy Galois Connections

The concept of Galois connection between power sets is generalized from the point of view of fuzzy logic. Studied is the case where the structure of truth values forms a complete residuated lattice. It is proved that fuzzy Galois connections are in one-to-one correspondence with binary fuzzy relations. A representation of fuzzy Galois connections by (classical) Galois connections is provided.     

A relative Shafarevich theorem

Suppose given a Galois étale cover YX of proper non-singular curves over an algebraically closed field k of prime characteristic p. Let H be its Galois group, and G any finite extension of H by a p-group P. We give necessary and sufficient conditions on G to be the Galois group of an étale cover of X dominating YX.in final form: 16 September 2003     

Obstructions locales au relèvement de revêtements galoisiens de courbes lisses

We give local obstructions to the possibility of lifting in zero characteristic a Galois cover of smooth algebraic curves defined over an algebraically closed field of characteristic p > 0.     

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties.In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group G^Mis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ? 2. A Galois cover f :X ? Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pal]. In this paper we prove two results about abelian covers:first, that if the building data are sufficiently ample, then the natural deformations surject on the Kuranishi family of X; second, that if the building data are sufficiently ample and generic, then Aut(X)= G.     

Algebraic function fields of one variable over finite fields are stable

It is shown that any algebraic curveC over a finite field has a separable cover of some degreen over the projective lineP ₁ such that the geometric Galois group of the Galois hull ofC |P ₁ is the full symmetric groupS _n. This work was partially supported by a grant from G.I.F. (German Israeli Foundation for Scientific Research and Development).     

On the number of Galois points for a plane curve in positive characteristic,II

For a smooth plane curve , we call a point a Galois point if the point projection at P is a Galois covering. We study Galois points in positive characteristic. We give a complete classification of the Galois group given by a Galois point and estimate the number of Galois points for C in most cases.      

Mixed degree number field computations

We present a method for computing complete lists of number fields in cases where the Galois group, as an abstract group, appears as a Galois group in smaller degree. We apply this method to find the 25 octic fields with Galois group \({{\mathrm{PSL}}}_2(7)\) and smallest absolute discriminant. We carry out a number of related computations, including determining the octic field with Galois group \(2^3{:}{{\mathrm{GL}}}_3(2)\) of smallest absolute discriminant.     

Universal abelian covers of quotient-cusps

 The quotient-cusp singularities are isolated complex surface singularities that are double-covered by cusp singularities. We show that the universal abelian cover of such a singularity, branched only at the singular point, is a complete intersection cusp singularity of embedding dimension 4. This supports a general conjecture that we make about the universal abelian cover of a ℚ-Gorenstein singularity. Received: 3 February 2001 / Revised version: 8 March 2002 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 14B05, 14J17, 32S25 This research was supported by grants from the Australian Research Council and the NSF (first author) and the the NSA (second author).     

Galois module structure of unramified covers

We use the theory of n-cubic structures to study the Galois module structure of the coherent cohomology groups of unramified Galois covers of varieties over the integers. Assuming that all the Sylow subgroups of the covering group are abelian, we show that the invariant that measures the obstruction to the existence of a “virtual normal integral basis” is annihilated by a product of certain Bernoulli numbers with orders of even K-groups of Z. We also show that the existence of such a basis is closely connected to the truth of the Kummer-Vandiver conjecture for the prime divisors of the degree of the cover. Partially supported by NSF grants # DMS05-01049 and # DMS01-11298 (via the Institute for Advanced Study).     

Solvable absolute Galois groups are metabelian

We show that solvable absolute Galois groups have an abelian normal subgroup such that the quotient is the direct product of two finite cyclic and a torsion-free procyclic group. In particular, solvable absolute Galois groups are metabelian. Moreover, any field with solvable absolute Galois group G admits a non-trivial henselian valuation, unless each Sylow-subgroup of G is either procyclic or isomorphic to Z ₂⋊Z/2Z. A complete classification of solvable absolute Galois groups (up to isomorphism) is given. Oblatum 22-IV-1998 & 1-IX-2000?Published online: 30 October 2000