Polarizations on Abelian Varieties and Self-dual ℓ-adic Representations of Inertia Groups

It is well known that every finite subgroup of GL _d (Q ) is conjugate to a subgroup of GL _d (Z ). However, this does not remain true if we replace general linear groups by symplectic groups. We say that G is a group of inertia type of G is a finite group which has a normal Sylow-p-subgroup with cyclic quotient. We show that if >d+1, and G is a subgroup of Sp _2d (Q ) of inertia type, then G is conjugate in GL _2d (Q ) to a subgroup of Sp _2d (Z ). We give examples which show that the bound is sharp. We apply these results to construct, for every odd prime , isogeny classes of Abelian varieties all of whose polarizations have degree divisible by ². We prove similar results for Euler characteristic of invertible sheaves on Abelian varieties over fields of positive characteristic.     

A Sylvester–Gallai Type Theorem for Abelian Groups

Mathematical Notes - This paper provides an elementary proof of inequalities previously obtained by the operator method and having applications in additive combinatorics. The method of proof allows...     

On the Skitovich–Darmois Theorem for Compact Abelian Groups

Let X be a separable compact Abelian group, Aut(X) the group of topological automorphisms of X, f _n: XX a homomorphism f _n(x)=nx, and X ⁽ⁿ⁾=Im f _n. Denote by I(X) the set of idempotent distributions on X and by (X) the set of Gaussian distributions on X. Consider linear statistics L ₁= ₁( ₁)+ ₂( ₂) and L ₂= ₁( ₁)+ ₂( ₂), where _j are independent random variables taking on values in X and with distributions _j, and _j, _jAut(X). The following results are obtained. Let X be a totally disconnected group. Then the independence of L ₁ and L ₂ implies that ₁, ₂I(X) if and only if X possesses the property: for each prime p the factor-group X/X ^(p) is finite. If X is connected, then there exist independent random variables _j taking on values in X and with distributions _j, and _j, _jAut(X) such that L ₁ and L ₂ are independent, whereas ₁, ₂(X) * I(X).     

Holomorphic Vector Bundle on Hopf Manifolds with Abelian Fundamental Groups

Let X be a Hopf manifolds with an Abelian fundamental group.E is a holomorphic vectorbundle of rank r with trivial pull-back to W=C~n-{0}.We prove the existence of a non-vanishingsection of L(?)E for some line bundle on X and study the vector bundles filtration structure of E.These generalize the results of D.Mall about structure theorem of such a vector bundle E.     

Note on Magic Squares and Magic Cubes on Abelian Groups

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Vanishing of Tate Homology — An Application of Stable Homology for Complexes

It has been proved that the vanishing of Tate homology is a sufficient condition for the derived depth formula to hold in [J. Pure Appl. Algebra, 219, 464–481(2015)]. In this paper, we investigate when Tate homology vanishes by studying the stable homology theory for complexes. Properties such as the balancedness and vanishing of stable homology for complexes are studied. Our results show that the vanishing of this homology can detect finiteness of homological dimensions of complexes and regularness of rings.     

On C.R. Rao’s Theorem for Locally Compact Abelian Groups

Doklady Mathematics - Let ξ 1, ξ2, ξ3 be independent random variables with values in a locally compact Abelian group X with nonvanishing characteristic functions, and aj, bj be...     

Mean‐Reverting Market Model: Speculative Opportunities and Non‐Arbitrage

The paper studies arbitrage opportunities and possible speculative opportunities for diffusion mean‐reverting market models. It is shown that the Novikov condition is satisfied for any time interval and for any set of parameters. It is non‐trivial because the appreciation rate has Gaussian distribution converging to a stationary limit. It follows that the mean‐reverting model is arbitrage‐free for any finite time interval. Further, it is shown that this model still allows some speculative opportunities: a gain for a wide enough set of expected utilities can be achieved for a strategy that does not require any hypothesis on market parameters and does not use estimation of these parameters.     

Subgroups of Finite Symmetric Groups

Recent results about primitive permutation groups are surveyed.     

Nontriviality of ω－Homology in Higher Dimensions

Ｎｏｎｔｒｉｖｉａｌｉｔｙ　ｏｆ　ω－Ｈｏｍｏｌｏｇｙ　ｉｎ　Ｈｉｇｈｅｒ　ＤｉｍｅｎｓｉｏｎｓＮｏｎｔｒｉｖｉａｌｉｔｙｏｆω－ＨｏｍｏｌｏｇｙｉｎＨｉｇｈｅｒＤｉｍｅｎｓｉｏｎｓＰａｎＪｉａｎｚｈｏｎｇ；ＳｈｅｎＸｉｎｙａｏ（Ｉｎｓｔｉｔｕｔｅｏ...     

Periodic Cyclic Homology of Iwahori–Hecke Algebras

We determine the periodic cyclic homology of the Iwahori–Hecke algebras H _q , for q * not a proper root of unity. (In this paper, by a proper root of unity we shall mean a root of unity other than 1.) Our method is based on a general result on periodic cyclic homology, which states that a weakly spectrum preserving morphism of finite type algebras induces an isomorphism in periodic cyclic homology. The concept of a weakly spectrum preserving morphism is defined in this paper, and most of our work is devoted to understanding this class of morphisms. Results of Kazhdan and Lusztig and Lusztig show that, for the indicated values of q, there exists a weakly spectrum preserving morphism _q : H _q J, to a fixed finite type algebra J. This proves that _q induces an isomorphism in periodic cyclic homology and, in particular, that all algebras H _q have the same periodic cyclic homology, for the indicated values of q. The periodic cyclic homology groups of the algebra H ₁ can then be determined directly, using results of Karoubi and Burghelea, because it is the group algebra of an extended affine Weyl group.     

Abelianness of Mumford–Tate Groups Associated to Some Unitary Groups

In this paper, we investigate the action of the -cohomology of the compact dual of a compact Shimura Variety S() on the -cohomology of S()> under a cup product. We use this to split the cohomology of S() into a direct sum of (not necessarily irreducible) -Hodge structures. As an application, we prove that for the class of arithmetic subgroups of the unitary groups U(p,q) arising from Hermitian forms over CM fields, the Mumford–Tate groups associated to certain holomorphic cohomology classes on S() are Abelian. As another application, we show that all classes of Hodge type (1,1) in H² of unitary four-folds associated to the group U(2,2) are algebraic.     

On Approximate Additive–Quartic and Quadratic–Cubic Functional Equations in Two Variables on Abelian Groups

We establish the generalized Hyers–Ulam stability of the additive–quartic and quadratic–cubic mappings with two variables on abelian groups.     

Hochschild Homology Invariants of Külshammer Type of Derived Categories

For a perfect field k of characteristic p > 0 and for a finite dimensional symmetric k-algebra A Külshammer studied a sequence of ideals of the centre of A using the p-power map on degree 0 Hochschild homology. In joint work with Bessenrodt and Holm we removed the condition to be symmetric by passing through the trivial extension algebra. If A is symmetric, then the dual to the Külshammer ideal structure was generalised to higher Hochschild homology in earlier work [23 Zimmermann , A. ( 2007 ). Fine Hochschild invariants of derived categories for symmetric algebras . Journal of Algebra 308 : 350 – 367 . [Google Scholar]]. In the present article we follow this program and propose an analogue of the dual to the Külshammer ideal structure on the degree m Hochschild homology theory also to not necessarily symmetric algebras.     

On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group,π_e(G) denotes the set of orders of elements in G.If Ω is a subsetof the set of natural numbers,h(Ω) stands for the number of isomorphism classes of finite groups withthe stone set Ω of element orders.We say that G is k-distinguishable if h(π_e(G))=k<∞,otherwiseG is called non-distinguishable.Usually,a 1-distinguishable group is called a characterizable group.Itis shown that if M is a sporadic simple group different from M_(12),M_(22),J_2,He,Suz,M~cL and O'N,then Aut(M) is characterizable by its element orders.It is also proved that if M is isomorphic toM_(12),M_(22),He,Suz or O'N,then h(π_e(Aut(M)))∈{1,∞}.     

Soluble Block-transitive Automorphism Groups of Design

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Equicentralizer Subgroups of Sporadic Simple Groups

A subgroup H of a group C is called an equicentralizer subgroup of G if thecentralizers of its nonidentity elements are all equal. We discuss the equicentrali-zer subgroups of sporadic simple groups. Using the orders of centralizers of theelements and the structure of the maximal subgroups, we prove that the equicent-     

On Direct Product of HX Groups

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Vertices of π-Irreducible Characters of Groups of Odd Order

If G is a group of odd order and χ ∈ Irr(G) lifts an irreducible Brauer character ?, then we associate to χ a canonical pair (Q, δ) up to G-conjugacy, where Q is a vertex of ? and δ ∈ Irr(Q) is a linear character of Q. We show that (Q, δ) is a Navarro vertex for χ. We also discuss examples.