Abelian varieties with finite abelian group action

We show that every action of a finite dihedral group on a closed orientable surface F \mathcal F extends to a 3-dimensional handlebody V \mathcal V , with ∂ V = F \partial \mathcal V=\mathcal F . In the case of a finite abelian group G G , we give necessary and sufficient conditions for a G G -action on a surface to extend to a compact 3 3 -manifold, or, equivalently in this case, to a 3-dimensional handlebody; in particular all (fixed-point) free actions of finite abelian groups extend to handlebodies. This is no longer true for free actions of arbitrary finite groups: we give a procedure which allows us to construct free actions of finite groups on surfaces which do not extend to a handlebody. We also show that the unique Hurwitz action of order 84 ( g − 1 ) 84(g-1) of P S L ( 2 , 27 ) PSL(2,27) on a surface F \mathcal F of genus g = 118 g=118 does not extend to any compact 3-manifold M M with ∂ M = F \partial M=\mathcal F , thus resolving the only case of Hurwitz actions of type P S L ( 2 , q ) PSL(2,q) of low order which remained open in an earlier paper (Math. Proc. Cambridge Philos. Soc. 117 (1995), 137–151).

Modular invariance property of association schemes is recalled in connection with our joint work with François Jaeger. Then we survey codes over F 2 discussing how codes, through their (various kinds of) weight enumerators, are related to (various kinds of) modular forms through polynomial invariants of certain finite group actions and theta series. Recently, not only codes over an arbitrary finite field but also codes over finite rings and finite abelian groups are considered and have been studied extensively. We show how the determination of the solutions of the modular invariance property of finite abelian groups (our joint work with Jaeger) is used to define the concept of Type II codes over arbitrary finite abelian groups. As an example of the usefulness of such Type II codes, we give an application to hermitian modular forms.

The fixed-point algebra of tensor-product actions of finite Abelian groups on UHF-algebras

Free actions of finite abelian groups on handlebodies

Perfect nonlinear functions between two finite abelian groups were studied by Carlet and Ding (J Complex 20:205---244, 2004) and Pott (Discret Math Appl 138:177---193, 2004), which can be regarded as a generalization of bent functions on finite abelian groups studied by Logachev et al. (Discret Math Appl 7:547---564, 1997). Poinsot (J Discret Math Sci Cryptogr 9:349---364, 2006), (Cryptogr Commun 4:1---23, 2012) extended this research to arbitrary finite groups, and characterized bent functions on finite nonabelian groups as well as perfect nonlinear functions between two arbitrary finite groups by the Fourier transforms of the related functions at irreducible unitary representations. The purpose of this paper is to study the characterizations of the bentness (perfect nonlinearity) of functions on arbitrary finite groups by the Fourier transforms of the related functions at irreducible characters. We will also give a characterization of a perfect nonlinear function by the relative pseudo-difference family.

We now deal with problems of p odd similar to those of Chapter V for p = 2. We lead off in section 43 with differentiable, orientation preserving actions of (Z p ) k , p an odd prime, on closed oriented manifolds V n . The primary aim is to give existence theorems for stationary points of such actions. Our interest in such problems has been aroused by the work of Borel [6, 9], although we attack the problem from a different point of view. An example of a corollary of our results is that if V n has one of its Pontryagin numbers not divisible by p then the action has a stationary point. We go on to note that if a toral group acts on V n without stationary points then [V n ] represents a torsion element of Ω n .

Transitive actions of finite abelian groups of sup-norm isometries

Let \Lambda be a finite abelian group. A dynamical system with transformation group \Lambda is a triple (A,\Lambda,\alpha) , consisting of a unital locally convex algebra A , the finite abelian group \Lambda and a group homomorphism \alpha:\Lambda\rightarrow\operatorname{Aut}(A) , which induces an action of \Lambda on A . In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal bundles with finite abelian structure group based on such dynamical systems.

We show that every finite abelian group occurs as the group of rational points of an ordinary abelian variety over F 2 \mathbb {F}_2 , F 3 \mathbb {F}_3 and F 5 \mathbb {F}_5 . We produce partial results for abelian varieties over a general finite field F q \mathbb {F}_q . In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over F q \mathbb {F}_q when q q is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over F 2 \mathbb {F}_2 .

In this paper, we survey some recent results on actions of finite groups on topological manifolds. Given an action of a finite group [Formula: see text] on a manifold [Formula: see text], these results provide information on the restriction of the action to a subgroup of [Formula: see text] of index bounded above by a number depending only on [Formula: see text]. Some of these results refer to the algebraic structure of the group, such as being abelian or nilpotent or admitting a generating subset of controlled size; other results refer to the geometry of the action, e.g. to the existence of fixed points, to the collection of stabilizer subgroups or to the action on cohomology.

group of odd order has only inner automorphisms. The group of automorphisms of the holomorph of an arbitrary group was studied by Gol'fand [3] who found some cases in which the outer automorphism group has order one or two. In the present paper I determine explicitly the outer automorphism group 0 of the holomorph H of an arbitrary finite abelian group G. If G is the direct product of a group of odd order, a group of order two, and a cyclic group of order 2n where n> 2, then 0 is the direct product of a finite number of groups of order two and a non-abelian group 0* of order six or eight. If n =2 then 0 * is isomorphic to the symmetric group of order six, and if n >3 then 0* is the octic group. In all other cases 0 is either trivial or the direct product of a finite number of groups of order two. Let A be the group of all automorphisms of the finite abelian group G, let GB be the group of all automorphisms of H that map G onto itself, and let g be the group of all inner automorphisms of H. Then (B3/J can be identified with the first cohomology group H'(A, G). Thus H'(A, G) can be regarded as a subgroup of 0. Now G is an ilnvariant subgroup of H, and it is known [6] that H has at most four invariant subgroups isomorphic to G. It follows that H'(A, G) has index at most four in 0.

Computing sparse Fourier sum of squares on finite abelian groups in quasi-linear time

A new framework for identifying absolute maximum nonlinear functions

A ‐tuple of core‐free subgroups of a finite group is said to be regular if has a regular orbit on the Cartesian product . The regularity number of , denoted by , is the smallest positive integer with the property that every such ‐tuple is regular. In this paper, we develop some general methods for studying the regularity of subgroup tuples in arbitrary finite groups, and we determine the precise regularity number of all almost simple groups with an alternating or sporadic socle. For example, we prove that and . We also formulate and investigate natural generalisations of several well‐studied problems on base sizes for finite permutation groups, including conjectures due to Cameron, Pyber and Vdovin. For instance, we extend earlier work of Burness, O'Brien and Wilson by proving that for every almost simple sporadic group, with equality if and only if is the Mathieu group . We also show that every triple of soluble subgroups in an almost simple sporadic group is regular, which generalises recent work of Burness on base sizes for transitive actions of sporadic groups with soluble point stabilisers.

This thesis presents some contributions in additive combinatorics and arithmetic Ramsey theory. More specifically, it deals with the interaction between combinatorics, number theory and additive combinatorics. This area saw a great improvement with the Szemerédi Regularity Lemma and some of the results that followed. The Regularity Lemma and its consequences have become a widely used tool in graph theory, combinatorics and number theory. Furthermore, its language and point of view has deeply changed the face of additive number theory, a fact universally acknowledged by the Abel award given to Szemerédi in 2012. One of the main reasons for the prize has been Szemerédi's theorem, a result regarding the existence of arbitrarily long arithmetic progressions in dense sets of the integers, the proof of which uses the Regularity Lemma in a key step. One of the earlier consequences of the Regularity Lemma was the Removal Lemma for graphs that was used by Ruzsa and Szemerédi to show Roth theorem, regarding the existence of 3-term arithmetic progressions in dense sets of the integers, in a combinatorial way. The Removal Lemma states that in any graph K with few copies of a subgraph, say a triangle, we can remove few edges from K so that the result contains no copy of the subgraph. This has become a key tool in the applications of the so-called Regularity Method, which has extensive literature in combinatorics, graph theory, number theory and computer science. In 2005 Green introduced a regularity lemma for Abelian groups as well as an algebraic removal lemma. The removal lemma for groups states that, for a given finite Abelian group G, if there are o(|G|^3) solution to x+y+z+t=0 with the variables taking values in S, a subset of G, then we can remove o(|G|) elements from S to make the set S solution-free. The main contributions of this work corresponds to extensions of the removal lemma for groups to either more general contexts, like non-necessary Abelian finite groups, or to linear systems of equations for finite Abelian groups. The main goal is to give a comprehensive and more general framework for many results in additive number theory like Szemerédi Theorem. In particular, we show that the removal lemma for groups by Green can be extended to non-necessary Abelian finite groups. Moreover, we prove a removal lemma for linear systems on finite fields: for every e>0 there exists a d>0 such that if A is a (k x m) linear system of equations with coefficients in a finite field F and the number of solutions to Ax=b, where each variable takes values from a subset Si in F is less than d times |F| raised to m-k, then by removing less than e|F| elements in each Si we can make the resulting sets solution-free, thus solving a conjecture by Green to that respect. Even more, if A is an integer linear system, G is a finite Abelian group, and the determinantal of A and |G| are coprime, then a similar statement holds. Let us mention that the last result allows us to characterize those linear systems where any set S with size proportional to G has a nontrivial solution in S, provided |G| is large enough. This extends the validity of Szmerédi's theorem to finite Abelian groups. These extensions of the removal lemma have been used in arithmetic Ramsey theory to obtain counting results for the number of monochromatic solutions of linear systems. The main result from a work by Frankl, Graham and Rödl in '88 states that the number of monochromatic solutions of regular systems in integer intervals is in fact a positive proportion of the total number of solutions. We give analogous results for solutions in Abelian groups with bounded exponent, for which the main tool in the torsion-free case cannot be applied. Density versions of these counting results are also obtained, in this case with a full characterization.