Construction techniques for Galois coverings of the affine line

For constructing un ramified coverings of the affine line in characteristicp, a general theorem about good reductions modulop of coverings of characteristic zero curves is proved. This is applied to modular curves to realize SL(2, ℤ/nℤ)/±1, with GCD(n, 6) = 1, as Galois groups of unramified coverings of the affine line in characteristicp, for p = 2 or 3. It is applied to the Klein curve to realize PSL(2, 7) for p = 2 or 3, and to the Macbeath curve to realize PSL(2, 8) for p = 3. By looking at curves with big automorphism groups, the projective special unitary groups PSU(3, p^v) and the projective special linear groups PSL(2, p^v) are realized for allp, and the Suzuki groups Sz(2^2v+1) are realized for p = 2. Jacobian varieties are used to realize certain extensions of realizable groups with abelian kernels.     

Hochschild （Co）homology of Galois Coverings of Grassmann Algebras

Let ∧ be the Z2-Galois covering of the Grassmann algebra A over a field k of characteristic not equal to 2. In this paper, the dimensional formulae of Hochschild homology and cohomology groups of ∧ are calculated explicitly. And the cyclic homology of∧ can also be calculated when the underlying field is of characteristic zero. As a result, we prove that there is an isomorphism from i≥1 HH^i（∧） to i≥1 HH^i（∧）.     

Mixed multiplier ideals and the irregularity of abelian coverings of smooth projective surfaces

A formula for the irregularity of abelian coverings of smooth projective surfaces is established. Explicit computations are performed and some applications are presented.     

Galois coverings and endomorphisms of projective varieties

We prove that the vector bundle associated to a Galois covering of projective manifolds is ample (resp. nef) under very mild conditions. This results is applied to the study of ramified endomorphisms of Fano manifolds with b ₂ = 1. It is conjectured that is the only Fano manifold admitting an endomorphism of degree d ≥ 2, and we verify this conjecture in several cases. An important ingredient is a generalization of a theorem of Andreatta–Wisniewski, characterizing projective space via the existence of an ample subsheaf in the tangent bundle. Marian Aprodu was supported in part by a Humboldt Research Fellowship and a Humboldt Return Fellowship. He expresses his special thanks to the Mathematical Institute of Bayreuth University for hospitality during the first stage of this work. Stefan Kebekus and Thomas Peternell were supported by the DFG-Schwerpunkt “Globale Methoden in der komplexen Geometrie” and the DFG-Forschergruppe “Classification of Algebraic Surfaces and Compact Complex Manifolds”. A part of this paper was worked out while Stefan Kebekus visited the Korea Institute for Advanced Study. He would like to thank Jun-Muk Hwang for the invitation.     

Galois Theory for the Selmer Group of an Abelian Variety

This paper concerns the Galois theoretic behavior of the p-primary subgroup Sel _A (F) _p of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the Galois group G(K/F) is a p-adic Lie group. Here p is any prime such that A has potentially good, ordinary reduction at all primes of F lying above p. The principal results concern the kernel and the cokernel of the natural map s _K/F Sel _A (F) _p Sel _A (K) _p ^G(K/F) where F is any finite extension of F contained in K. Under various hypotheses on the extension K/F, it is proved that the kernel and cokernel are finite. More precise results about their structure are also obtained. The results are generalizations of theorems of B.Mazurand M. Harris.     

On the Galois covers of degenerations of surfaces of minimal degree

We investigate the topological structures of Galois covers of surfaces of minimal degree (i.e., degree n) in ${\mathbb{CP}}^{n+1}$. We prove that for $n\ge 5$, the Galois covers of any surfaces of minimal degree are simply-connected surfaces of general type.     

On Galois Coverings of the Enveloping Algebras of Self-Injective Nakayama Algebras

Abstract

There is constructed a Galois covering F of the enveloping K-algebra A ^e of a self-injective Nakayama K-algebra A such that the right A ^e -module A is of the first kind with respect to F. Then, with the help of the constructed Galois covering, the Auslander-Reiten translation period of A is computed.     

Galois Extensions of Boolean Algebras

We characterize Galois extensions of Boolean algebras as finite extensions with the independent set of generators, answering a question of D. Monk.     

Galois Graphs: Walks, Trees and Automorphisms

Given a symmetric polynomial (x, y) over a perfect field k of characteristic zero, the Galois graph G() is defined by taking the algebraic closure as the vertex set and adjacencies corresponding to the zeroes of (x, y). Some graph properties of G(), such as lengths of walks, distances and cycles are described in terms of . Symmetry is also considered, relating the Galois group Gal( /k) to the automorphism group of certain classes of Galois graphs. Finally, an application concerning modular curves classifying pairs of isogeny elliptic curves is revisited.     

Inertia groups and abelian surfaces

This paper classifies the finite groups that occur as inertia groups associated to abelian surfaces. These groups can be viewed as Galois groups for the smallest totally ramified extension over which an abelian surface over a local field acquires semistable reduction. The results extend earlier elliptic curves results of Serre and Kraus.     

Modularity of abelian varieties over Q with bad reduction in one prime only

We show that certain abelian varieties over Q with bad reduction at one prime only are modular by using methods based on the tables of Odlyzko and class field theory.     

Enumerating typical abelian prime-fold coverings of a circulant graph

Enumerating the isomorphism classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory (see [R. Feng, J.H. Kwak, J. Kim, J. Lee, Isomorphism classes of concrete graph coverings, SIAM J. Discrete Math. 11 (1998) 265-272; R. Feng, J.H. Kwak, Typical circulant double coverings of a circulant graph, Discrete Math. 277 (2004) 73-85; R. Feng, J.H. Kwak, Y.S. Kwon, Enumerating typical circulant covering projections onto a circulant graph, SIAM J. Discrete Math. 19 (2005) 196-207; SIAM J. Discrete Math. 21 (2007) 548-550 (erratum); M. Hofmeister, Graph covering projections arising from finite vector spaces over finite fields, Discrete Math. 143 (1995) 87-97; M. Hofmeister, Enumeration of concrete regular covering projections, SIAM J. Discrete Math. 8 (1995) 51-61; M. Hofmeister, A note on counting connected graph covering projections, SIAM J. Discrete Math. 11 (1998) 286-292; J.H. Kwak, J. Chun, J. Lee, Enumeration of regular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11 (1998) 273-285; J.H. Kwak, J. Lee, Isomorphism classes of graph bundles, Canad. J. Math. XLII (1990) 747-761]). A covering is called abelian (or circulant, respectively) if its covering graph is a Cayley graph on an abelian (or a cyclic, respectively) group. A covering p from a Cayley graph onto another Cay (Q,Y) is called typical if the map p:A→Q on the vertex sets is a group epimorphism. Recently, the isomorphism classes of connected typical circulant r-fold coverings of a circulant graph are enumerated in [R. Feng, J.H. Kwak, Typical circulant double coverings of a circulant graph, Discrete Math. 277 (2004) 73-85] for r=2 and in [R. Feng, J.H. Kwak, Y.S. Kwon, Enumerating typical circulant covering projections onto a circulant graph, SIAM J. Discrete Math. 19 (2005) 196-207; SIAM J. Discrete Math. 21 (2007) 548-550 (erratum)] for any r. As a continuation of these works, we enumerate in this paper the isomorphism classes of typical abelian prime-fold coverings of a circulant graph.     

量子外代数的Galois覆盖代数的Hochschild上同调

本文研究一类量子代数$\Lambda^n_q$的Hochschild上同调.量子代数$\Lambda^n_q$的极小投射双模分解被构造, $\Lambda^n_q$的各阶Hochschild上同调群的维数被清晰的给出.此外,对一些特殊的情况, $\Lambda^n_q$的上同调环也被清晰的刻画.     

New coverings of t-sets with (t + 1)-sets

The minimum number of k-subsets out of a v-set such that each t-set is contained in at least one k-set is denoted by C(v, k, t). In this article, a computer search for finding good such covering designs, leading to new upper bounds on C(v, k, t), is considered. The search is facilitated by predetermining automorphisms of desired covering designs. A stochastic heuristic search (embedded in the general framework of tabu search) is then used to find appropriate sets of orbits. A table of upper bounds on C(v, t + 1, t) for v 28 and t 8 is given, and the new covering designs are listed. © 1999 John Wiley & Sons, Inc. J. Combin Designs 7: 217–226, 1999     

Defining equations of the universal abelian surfaces with level three structure

In this paper we write down the defining equations of abelian surface with level three structures explicitly, and we see that their coefficients are given by Siegel modular forms of level three with some characters.     

A Galois correspondence for countable short recursively saturated models of PA

In this paper we investigate the properties of automorphism groups of countable short recursively saturated models of arithmetic. In particular, we show that Kaye's Theorem concerning the closed normal subgroups of automorphism groups of countable recursively saturated models of arithmetic applies to automorphism groups of countable short recursively saturated models as well. That is, the closed normal subgroups of the automorphism group of a countable short recursively saturated model of PA are exactly the stabilizers of the invariant cuts of the model which are closed under exponentiation. This Galois correspondence is used to show that there are countable short recursively saturated models of arithmetic whose automorphism groups are not isomorphic as topological groups. Moreover, we show that the automorphism groups of countable short arithmetically saturated models of PA are not topologically isomorphic to the automorphism groups of countable short recursively saturated models of PA which are not short arithmetically saturated (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Seshadri constants in finite subgroups of abelian surfaces

Given an étale quotient q : X → Y of smooth projective varieties we relate the simple Seshadri constant of a line bundle M on Y with the multiple Seshadri constant of q ^* M in the points of the fiber. We apply this method to compute the Seshadri constant of polarized abelian surfaces in the points of a finite subgroup.      

Direct twisted Galois stratification

The theory ACFA admits a primitive recursive quantifier elimination procedure. It is therefore primitive recursively decidable.     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.