On the Brauer group of the product of a torus and a semisimple algebraic group

Let T be a torus (not assumed to be split) over a field F, and denote by nH _et ² (X,{ie375-1}) the subgroup of elements of the exponent dividing n in the cohomological Brauer group of a scheme X over the field F. We provide conditions on X and n for which the pull-back homomorphism nH _et ² (T,{ie375-2}) → _n H _et ² (X × _F T, {ie375-3}) is an isomorphism. We apply this to compute the Brauer group of some reductive groups and of non-singular affine quadrics. Apart from this, we investigate the p-torsion of the Azumaya algebra defined Brauer group of a regular affine scheme over a field F of characteristic p > 0.     

On a pairing for algebraic tori

Let T be an algebraic torus over a field F. There is a pairing between the groups of torsors for the torus T and its dual with values in the third Galois cohomology group over all field extensions of F. We study the kernel of this pairing.     

Extension of finite solvable torsors over a curve

Let R be a complete discrete valuation ring with fraction field K and with algebraically closed residue field of positive characteristic p. Let X be a smooth fibered surface over R. Let G be a finite, étale and solvable K-group scheme and assume that either |G| = p ⁿ or G has a normal series of length 2. We prove that for every connected and pointed G-torsor Y over the generic fibre ${X_{\eta}}$ of X there exist a regular fibered surface ${\widetilde{X}}$ over R and a model map ${\widetilde{X}\to X}$ such that Y can be extended to a torsor over ${\widetilde{X}}$ possibly after extending scalars.     

Additive operators preserving rank-additivity on symmetry matrix spaces

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS _n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F ^*, thenT is an additive injective operator preserving rank-additivity onS _n(F) if and only if there exists an invertible matrixU∈M _n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ^?U^T, ?X=(x_ij)∈S_n(F) wherec∈F ^*,X ^?=(?(x _ij)). As applications, we determine the additive operators preserving minus-order onS _n(F) over the fieldF.     

On the subfields of cyclotomic function fields

Let K = F(T) be the rational function field over a finite field of q elements. For any polynomial f(T) ∈ F [T] with positive degree, denote by Λ _f the torsion points of the Carlitz module for the polynomial ring F[T]. In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield M of the cyclotomic function field K(Λ _P ) of degree k over F(T), where P ∈ F[T] is an irreducible polynomial of positive degree and k > 1 is a positive divisor of q ? 1. A formula for the analytic class number for the maximal real subfield M ⁺ of M is also presented. Futhermore, a relative class number formula for ideal class group of M will be given in terms of Artin L-function in this paper.     

Pairs of alternating forms and products of two skew-symmetric matrices

Let k be a field, and let S,T,S₁,T₁ be skew-symmetric matrices over k with S,S₁ both nonsingular (if k has characteristic 2, a skew-symmetric matrix is a symmetric one with zero diagonal). It is shown that there exists a nonsingular matrix P over k with P'SP = S₁, P'TP = T₁ (where P' denotes the transpose of P) if and only if S^-1T and S^-1₁T₁ are similar. It is also shown that a 2m×2m matrix over k can be factored as ST, with S,T skew-symmetric and S nonsingular, if and only if A is similar to a matrix direct sum B⊕B where B is an m×m matrix over k. This is equivalent to saying that all elementary divisors of A occur with even multiplicity. An extension of this result giving necessary and sufficient conditions for a square matrix to be so expressible, without assuming that either S or T is nonsingular, is included.     

Darboux transforms and spectral curves of Hamiltonian stationary Lagrangian tori

The multiplier spectral curve of a conformal torus f : T ² → S ⁴ in the 4-sphere is essentially (Bohle et al., Conformal maps from a 2-torus to the 4-sphere. arXiv:0712.2311) given by all Darboux transforms of f. In the particular case when the conformal immersion is a Hamiltonian stationary torus ${f: T^2 \to\mathbb{R}^4}$ in Euclidean 4-space, the left normal N : M → S ² of f is harmonic, hence we can associate a second Riemann surface: the eigenline spectral curve of N, as defined in Hitchin (J Differ Geom 31(3):627–710, 1990). We show that the multiplier spectral curve of a Hamiltonian stationary torus and the eigenline spectral curve of its left normal are biholomorphic Riemann surfaces of genus zero. Moreover, we prove that all Darboux transforms, which arise from generic points on the spectral curve, are Hamiltonian stationary whereas we also provide examples of Darboux transforms which are not even Lagrangian.     

Geometry of compact complex manifolds with maximal torus action

We study the geometry of compact complex manifolds M equipped with a maximal action of a torus T = (S ¹) ^k . We present two equivalent constructions that allow one to build any such manifold on the basis of special combinatorial data given by a simplicial fan Σ and a complex subgroup H ? T _? = (?*) ^k . On every manifold M we define a canonical holomorphic foliation F and, under additional restrictions on the combinatorial data, construct a transverse Kähler form ω _F . As an application of these constructions, we extend some results on the geometry of moment-angle manifolds to the case of manifolds M.     

Construction of vector lists and isomorph rejection

It is observed that if Δ is a system of orbit representatives for the action of a finite group G on a G-stable subset L of a finite set S and if F is a field of characteristic zero, then F^S is an algebra of dimension |S|. Furthermore, if S = R^D, the set of functions from a finite set D to a finite set R, then F^S has a multilinear structure. A general problem is stated: Given linear operators T₁ and T₂, construct vectors v₁, … v_t?F^S such that T₂I_Δ = T₁(v₁ + … + v_t) where l_Δ is the indicator or characteristic function of Δ. (Note that this construction gives, in some sense, a solution to the problem of isomorph rejection.) For appropriate choices of T₁ and T₂, two approaches to this construction problem are considered. These are the principle of inclusion-exclusion and backtrack computer programming. In particular, these approaches are discussed when S = R^D and the vectors to be constructed are pure or homogeneous tensors.     

Traces on the skein algebra of the torus

For a surface F, the Kauffman bracket skein module of F×[0,1], denoted K(F), admits a natural multiplication which makes it an algebra. When specialized at a complex number t, nonzero and not a root of unity, we have Kt(F), a vector space over C. In this paper, we will use the product-to-sum formula of Frohman and Gelca to show that the vector space Kt(T²) has five distinct traces. One trace, the Yang-Mills measure, is obtained by picking off the coefficient of the empty skein. The other four traces on Kt(T²) correspond to the four singular points of the moduli space of flat SU(2)-connections on the torus.     

Congruence and Conjunctivity of Matrices

Congruence of arbitrary square matrices over an arbitrary field is treated here by elementary classical methods, and likewise for conjunctivity of arbitrary square matrices over an arbitrary field with involution. Uniqueness results are emphasized, since they are largely neglected in the literature. In particular, it is shown that a matrix S is congruent [conjunctive] to S₀⊕S₁ with S₁ nonsingular, and that if S₁ here is of maximal size among all nonsingular matrices R₁ for which R₀⊕R₁ is congruent [conjunctive] to S, then the congruence [conjunctivity] class of S determines that of S₁. Partially canonical forms (most of them already known) are derived, to the extent that they do not depend on the field. Nearly canonical forms are derived for “neutral” matrices (those congruent or conjunctive with block matrices $\text{O}\text{N}\text{M}\text{O}$ with the two zero blocks being square). For a neutral matrix S over a field F,the F-congruence [F-conjunctivity] class of S is determined by the F-equivalence class of the pencil S+tS' [S+tS¹] and, if the pencil is nonsingular, by the F[t]-equivalence class of S+tS' [S+tS¹].     

Simultaneous orderings in function fields

For every Dedekind domain R, Bhargava defined the factorials of a subset S of R by introducing the notion of p-ordering of S, for every maximal ideal p of R. We study the existence of simultaneous ordering in the case S=R=O_K, where O_K is the ring of integers of a function field K over a finite field F_q. We show, that when O_K is the ring of integers of an imaginary quadratic extension K of F_q(T), K=F_q(T)/(Y²-D(T)), then there exists a simultaneous ordering if and only if degD?1.     

Epimorphisms of knot groups onto free products

Let k be a knot in S³. There is an epimorphism from π₁(S³−k) onto a free product of two nontrivial cyclic groups sending a meridian to an element of length two iff k has property Q (Topology of Manifolds, Markham, Chicago, IL, 1970, pp. 195-199) that is if there is a closed surface F in S³ containing k, such that k is imprimitive in H₁(X) and in H₁(Y) where X and Y are the closures of the components of S³−F. We give answers to questions of Simon (1970) about properties Q, Q∗ and Q∗∗. Epimorphisms from knot groups onto torus knot groups are also studied and some results on property P and surgery are included.     

On Selmer groups of abelian varieties over ℓ-adic Lie extensions of global function fields

Let F be a global function field of characteristic p > 0 and A/F an abelian variety. Let K/F be an ?-adic Lie extension (? ≠ p) unramified outside a finite set of primes S and such that Gal(K/F) has no elements of order ?. We shall prove that, under certain conditions, Sel _A (K) _? ^∨ has no nontrivial pseudo-null submodule.     

Degeneration of torsors over families of del Pezzo surfaces

Let S be a split family of del Pezzo surfaces over a discrete valuation ring such that the general fiber is smooth and the special fiber has $\mathbf{ADE}$-singularities. Let G be the reductive group given by the root system of these singularities. We construct a G-torsor over S whose restriction to the generic fiber is the extension of structure group of the universal torsor. This extends a construction of Friedman and Morgan for individual singular del Pezzo surfaces. In case of very good residue characteristic, this torsor is unique and infinitesimally rigid.     

Equivariant Total Ring of Fractions and Factoriality of Rings Generated by Semi-Invariants

Let F be an affine flat group scheme over a commutative ring R, and S an F-algebra (an R-algebra on which F acts). We define an equivariant analogue Q _F (S) of the total ring of fractions Q(S) of S. It is the largest F-algebra T such that S ? T ? Q(S), and S is an F-subalgebra of T. We study some basic properties.

Utilizing this machinery, we give some new criteria for factoriality (unique factorization domain property) of (semi-)invariant subrings under the action of affine algebraic groups, generalizing a result of Popov. We also prove some variations of classical results on factoriality of (semi-)invariant subrings. Some results over an algebraically closed base field are generalized to those over an arbitrary base field.     

Galois cohomology of the classical groups over imperfect fields

Elaborating on techniques of Bayer-Fluckiger and Parimala, we prove the following strong version of Serre’s Conjecture II for classical groups: let G be a simply connected absolutely simple group of outer type A_n or of type B_n, C_n or D_n (non trialitarian) defined over an arbitrary field F. If the separable dimension of F is at most 2 for every torsion prime of G, then every G-torsor is trivial.     

Products of idempotent matrices

For some years it has been known that every singular square matrix over an arbitrary field F is a product of idempotent matrices over F. This paper quantifies that result to some extent. Main result: for every field F and every pair (n,k) of positive integers, an n×n matrix S over F is a product of k idempotent matrices over F iff rank(I ? S)?k· nullity S. The proof of the “if” part involves only elementary matrix operations and may thus be regarded as constructive. Corollary: (for every field F and every positive integer n) each singular n×n matrix over F is a product of n idempotent matrices over F, and there is a singular n×n matrix over F which is not a product of n ? 1 idempotent matrices.