Existence of abelian group code partitions

We prove certain results pertaining to some isomorphism properties of commutative modular group algebras and briefly review a paper by pointing out some obvious mistakes and essential incorrectness.

Applications to cryptography of twisting commutative algebraic groups

Let G be an abelian group and let R be a commutative ring with identity. Denote by R t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ℬ(R) and ℬ(R t G) the nil radicals of R and R t G, respectively, by G p the p-component of G and by G 0 the torsion subgroup of G. We prove that: i. If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ℬ(R) = 0, then there exists a twisted group algebra R t 1 (G/G p ) such that R t G/ℬ(R t G) ≅ R t 1 (G/G p ) as R-algebras; ii. If R is a ring of prime characterisitic p and R* is p-divisible, then ℬ(R t G) = 0 if and only if ℬ(R) = 0 and G p = 1; and iii. If B(R) = 0, the orders of the elements of G 0 are not zero divisors in R, H is any group and the commutative twisted group algebra R t G is isomorphic as R-algebra to some twisted group algebra R t 1 H, then R t G 0 ≅ R t 1 H 0 as R-algebras.

A deformation-theoretic version of Maschke's theorem for modular group algebras: The commutative case

The subject of algebraic groups has had a rapid development in recent years. Leaving aside the late research by many people on the Albanese and Picard variety, it has received much substance and impetus from the work of Severi on commutative algebraic groups over the complex number field, that of Kolchin, Chevalley, and Borel on algebraic groups of matrices, and especially Weil's research on abelian varieties and algebraic transformation spaces. The main purpose of the present paper is to give a more or less systematic account of a large part of what is now known about general algebraic groups, which may be abelian varieties, algebraic groups of matrices, or actually of neither of these types.

We establish new measures of linear independence of logarithms on a commutative algebraic group. Let G be a connected commutative algebraic group over Q−− and let tG be the tangent space at the origin. We consider a vector u∈tG⊗Q−C such that its image by the exponential map of the Lie group G(C) is an algebraic point p∈G(Q−−). Let V be a hyperplane in tG. We obtain lower bounds for the distance d(u,V) between u and V⊗Q−C in the rational case, where V=tH is the tangent space at the origin of an algebraic connected subgroup of G. These lower bounds are the best currently known in terms of the height h(p) of p. They generalize measures of linear forms in logarithms previously obtained by Gaudron. Our approach is based on new arguments which allow us to exclude the so-called periodic case in the demonstration, by revisiting previous work of Bertrand and Philippon. Our proofs also rely on tools from Bost’s slope theory of hermitian vector bundles. Moreover, we present ultrametric analogues of our results, and we deal with the case where V=tH is a linear subspace of any dimension.

This work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group, defined over the field of algebraic numbers $\overline{\mathbb{Q}}$ . Let G be such a group. Let W be a hyperplane of the tangent space at the origin of G, defined over $\overline{\mathbb{Q}}$ , and u be a complex point of this tangent space, such that the image of u by the exponential map of the Lie group G(ℂ) is an algebraic point. Then we obtain a lower bound for the distance between u and W⊗ℂ, which improves the results known before and which is, in particular, the best possible for the height of the hyperplane W. The proof rests on Baker’s method and Hirata’s reduction as well as a new arithmetic argument (Chudnovsky’s process of variable change) which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof.

Let k be a number field, let A be a commutative algebraic group defined over k and let n be a positive integer. We will prove that for almost all primes p ∈ k, the existence of a counterexample to the localglobal divisibility by p in A assures the existence of a counterexample to the local-global divisibility by p in A, for all positive integers s. In particular, we will prove the existence of counterexamples to the localglobal divisibility by 2 and 3, for every n ≥ 2, in the case when k = Q and A is an elliptic curve. Furthermore, we will show a method to find numerical examples. We will use this method to produce numerical counterexamples to the local-global divisibility by 2, for every n ≥ 2, in elliptic curves defined over Q.

Let G be a commutative algebraic group embedded in projective space and \(\Gamma \) a finitely generated subgroup of G. From these data we construct a chain of algebraic subgroups of G which is intimately related to obstructions to multiplicity or interpolation estimates used in transcendental number theory and algebraic independence. Let \(\gamma _1,\ldots ,\gamma _l\) denote a family of generators of \(\Gamma \) and, for any \(S>1\), let \(\Gamma (S)\) be the set of elements \(n_1\gamma _1+\cdots +n_l\gamma _l\) with integers \(n_j\) such that \(|n_j| < S\). Then this chain of subgroups controls, for large values of S, the distribution of \(\Gamma (S)\) with respect to algebraic subgroups of G. As an application we essentially determine (up to multiplicative constants) the locus of common zeros of all \(P \in H^0(\overline{G} ,\mathcal{O}(D))\) which vanish to at least some given order at all points of \(\Gamma (S)\). When D is very small this result reduces to a multiplicity estimate; when D is very large it is a kind of interpolation estimate.

Families of translations of commutative algebraic groups

Mesure d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif

Two operators or two subgroups of a group will be called I-conjugate whenever they are conjugate under the group of isomorphisms of this group. It is well known that all the cyclic subgroups of highest order contained in any abelian group are I-conjugate. A necessary and sufficient condition that all the cyclic subgroups of any other order contained in an abelian group G of order pmm, p being a prime number, are I-conjugate is that all the invariants of G are equal to each other. When this condition is satisfied we shall prove, in particular, that the number of the subgroups of order pa, a c m , contained in G is equal to the number of its subgroups of index pa. It is known that the ?-subgroup of G is composed of the pth power of every operator of G. The concept of +-subgroup in connection with abelian groups can readily be extended by considering the characteristic subgroup of G composed of the path power of every operator of G. This subgroup may be called the Oa-subgroup of G. It is composed of all the operators of G which are found in each one of its subgroups of index pa, and all the operators of highest order in a 4.-subgroup of G are I-conjugate. In particular, the 0&-subgroup generated by operators of order p is the fundamental characteristic subgroup of G. t In fact, when p > 2 the 0a-subgroup which is generated by operators of order pr is contained in every characteristic subgroup of G which involves operators of order pr. The quotient group of G corresponding to a 4a-subgroup may be called a 0,fi-quotient group. This quotient group is simply isomorphic with the subgroup of G generated bv all its operators whose orders divide p> Hence the following: THEOREM. The number of the subgroups of index pa contained in any abelian

The notion of Reiter's Segal algebra in commutative group algebras is generalized to a notion of Segal algebra in more general classes of commutative Banach algebras. Then we introduce a family of Segal algebras in commutative Banach algebras under considerations and study some properties of them.

Let \mathbb K be an algebraically closed field of characteristic zero, \mathbb G_m=(\mathbb K\setminus{0},\times) be its multiplicative group, and \mathbb G_a=(\mathbb K,+) be its additive group. Consider a commutative linear algebraic group \mathbb G=(\mathbb G_m)^r\times\mathbb G_a . We study equivariant \mathbb G -embeddings, i.e. normal \mathbb G -varieties X containing \mathbb G as an open orbit. We prove that X is a toric variety and all such actions of \mathbb G on X correspond to Demazure roots of the fan of X . In these terms, the orbit structure of a \mathbb G -variety X is described.

Let K[?] denote the group algebra of an infinite locally finite group?. In recent years, the lattice of ideals of K[?] has been extensively studied under the assumption that? is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra K[V] stable under the action of an appropriate automorphism group of V. Specifically, in this paper, we let B be a quasi-simple group of Lie type defined over an infinite locally finite field F, and we let V be a finite-dimensional vector space over a field E of the same characteristic p. If B acts nontrivially on V by way of the homomorphism Φ: G → GL(V), and if V has no proper G-stable subgroups, then we show that the augmentation ideal ωK[V] is the unique proper G-stable ideal of K[V] when char K ¬= p. The proof of this result requires, among other things, that we study characteristic p division rings D, certain multiplicative subgroups G of D*, and the action of G on the group algebra K[A], where A is the additive group D + . In particular, properties of the quasi-simple group G come into play only in the final section of this paper.