Commutative Algebraic Groups Research Articles

Well-approximated points on linear extensions of elliptic curves

We employ a result on linear forms in logarithms of algebraic points on commutative algebraic groups, a study initiated by Philippon and Waldschmidt, a so-called “local nullstellen inequality” of Brownawell, and some elementary analytic estimates to study the approximation properties of coordinates of non-generic points on a linear (algebraic) group extension of an elliptic curve.

On algebraic integrability of Gelfand-Zeitlin fields

We generalize a result of Kostant and Wallach concerning the algebraic integrability of the Gelfand-Zeitlin vector fields to the full set of strongly regular elements in \( \mathfrak{g}\mathfrak{l} \)(n, ℂ). We use decomposition classes to stratify the strongly regular set by subvarieties \( {X_\mathcal{D}} \). We construct an etale cover \( {\hat{\mathfrak{g}}}_\mathcal{D} \) of \( {X_\mathcal{D}} \) and show that \( {X_\mathcal{D}} \) and \( {\hat{\mathfrak{g}}}_\mathcal{D} \) are smooth and irreducible. We then use Poisson geometry to lift the Gelfand-Zeitlin vector fields on \( {X_\mathcal{D}} \) to Hamiltonian vector fields on \( {\hat{\mathfrak{g}}}_\mathcal{D} \) and integrate these vector fields to an action of a connected, commutative algebraic group.

Invariant ideals of abelian group algebras under the torus action of a field, I

Let V = V 1 ⊕ V 2 be a finite-dimensional vector space over an infinite locally-finite field F. Then V admits the torus action of G = F • by defining ( v 1 ⊕ v 2 ) g = v 1 g − 1 ⊕ v 2 g . If K is a field of characteristic different from that of F, then G acts on the group algebra K [ V ] and it is an interesting problem to determine all G-stable ideals of this algebra. In this paper, we consider the special case when V 1 and V 2 are both 1-dimensional and we show that there are just four G-stable proper ideals of K [ V ] .

Complexity of multiplication in commutative group algebras over fields of prime characteristic

The objective of this research is to study the complexity of multiplication in commutative group algebras over arbitrary fields of prime characteristic. In order to solve this problem, we suggest a method to find the structure of group algebras which allows us to use the Alder–Strassen theorem to obtain lower bounds and the Bläser theorem describing all algebras of minimal rank to obtain upper bounds.

Modulus of a rational map into a commutative algebraic group

For a rational map $\phi: X \to G$ from a normal algebraic variety $X$ to a commutative algebraic group $G$, we define the modulus of $\phi$ as an effective divisor on $X$. We study the properties of the modulus. This work generalizes the known theories for curves to higher dimensional varieties.

Intersecting a plane with algebraic subgroups of multiplicative groups

Consider an arbitrary algebraic curve defined over the field of all alge- braic numbers and sitting in a multiplicative commutative algebraic group. In an earlier article from 1999 bearing almost the same title, we studied the intersection of the curve and the union of all algebraic subgroups of some fixed codimension. With codimension one the resulting set has bounded height properties, and with codimension two it has finiteness properties. The main aim of the present work is to make a start on such problems in higher dimension by proving the natural analogues for a linear surface (with codimensions two and three). These are in accordance with some general conjectures that we have recently proposed else- where.

Applications to cryptography of twisting commutative algebraic groups

We give an overview on twisting commutative algebraic groups and applications to discrete log-based cryptography. We explain how discrete log-based cryptography over extension fields can be reduced to cryptography in primitive subgroups. Primitive subgroups in turn are part of a general theory of tensor products of commutative algebraic groups and Galois modules (or twists of commutative algebraic groups), and this underlying mathematical theory can be used to shed light on discrete log-based cryptosystems. We give a number of concrete examples, to illustrate the definitions and results in an explicit way.

Generalized albanese and its dual

A generalization of the Albanese variety to the case of a singular projective variety X over an algebraically closed field k is given in [ESV], where H. Esnault, V. Srinivas and E. Viehweg constructed a universal regular quotient of the Chow group CH0(X)deg 0 of 0-cycles of degree 0 modulo rational equivalence. This is a smooth connected commutative algebraic group, universal for rational maps from X to smooth commutative algebraic groups which factor through a homomorphism of groups CH0(X)deg 0 ��! G(k). Suppose now that in addition k is of characteristic 0. Interpreting this algebraic group as a generalized 1-motive in the sense of Laumon [L], we may ask for the dual 1-motive. The intention of these notes was to describe the functor which is represented by the dual 1-motive. This forms the main result of this work. The notion of dual 1-motive allows to treat the problem in a more general way: We consider certain categories of rational maps from a projective variety to commutative algebraic groups (the category of rational maps factoring through CH0(X)deg 0 is a special case). A necessary and sufficient condition for the existence of an object of such a category satisfying the universal mapping property is given, as well as a construction of these universal objects via their dual 1-motives. In particular, this provides an independent proof of the existence and an explicit construction of the universal regular quotient for algebraically closed base field of characteristic 0.

On the Commutative Twisted Group Algebras

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.

Étude du cas rationnel de la théorie des formes linéaires de logarithmes

We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called rational case. More precisely, let k be a number field and v 0 be an arbitrary place of k. Let G be a commutative algebraic group defined over k and H be a connected algebraic subgroup of G. Denote by Lie ( H ) its Lie algebra at the origin. Let u ∈ Lie ( G ( C v 0 ) ) a logarithm of a point p ∈ G ( k ) . Assuming (essentially) that p is not a torsion point modulo proper connected algebraic subgroups of G, we obtain lower bounds for the distance from u to Lie ( H ) ⊗ k C v 0 . For the most part, they generalize the measures already known when G is a linear group. The main feature of these results is to provide a better dependence in the height log a of p, removing a polynomial term in log log a . The proof relies on sharp estimates of sizes of formal subschemes associated to H (in the sense of Bost) obtained from a lemma by Raynaud as well as an absolute Siegel lemma and, in the ultrametric case, a recent interpolation lemma by Roy.

Almost-triangular Hopf Algebras

In this paper, we consider a finite dimensional semisimple cosemisimple quasitriangular Hopf algebra \((H,R\,)\) with \(R^{\,21}R\in C(H\otimes H\,)\) (we call this type of Hopf algebras almost-quasitriangular) over an algebraically closed field \(k\). We denote by \(B\) the vector space generated by the left tensorand of \(R^{\,21}R\). Then \(B\) is a sub-Hopf algebra of \(H\). We proved that when \(\dim B\) is odd, \(H\) has a triangular structure and can be obtained from a group algebra by twisting its usual comultiplication [14]; when \(\dim B\) is even, \(H\) is an extension of an abelian group algebra and a triangular Hopf algebra, and may not be triangular. In general, an almost-triangular Hopf algebra can be viewed as a cocycle bicrossproduct.

Commutative Group Algebras of Summable p-Groups

Let G be an Abelian group whose p-component G p is summable and let F be a field of characteristic p. Let S(FG) be the normalized unit p-group of the group algebra FG. The main results of the present article are that G p is a direct factor of S(FG) with totally projective complement, provided G p is of countable length and F is perfect; and, in particular, under these circumstances S(FG) is summable if and only if G p is summable. Moreover, if FG≅ FH as F-algebras for some group H and if G p is summable, then H p is summablale. These achievements improve results due to Hill and Ullery (1997) and also their generalizations given by Danchev (2000ab20012004).

Twisting commutative algebraic groups

If V is a commutative algebraic group over a field k, O is a commutative ring that acts on V, and I is a finitely generated free O -module with a right action of the absolute Galois group of k, then there is a commutative algebraic group I ⊗ O V over k, which is a twist of a power of V. These group varieties have applications to cryptography (in the cases of abelian varieties and algebraic tori over finite fields) and to the arithmetic of abelian varieties over number fields. For purposes of such applications we devote this article to making explicit this tensor product construction and its basic properties.

When is a semilocal group algebra continuous?

It is shown that (i) a semilocal group algebra KG of an infinite nilpotent group G over a field K of characteristic p > 0 is CS (equivalently continuous) if and only if G = P × H , where P is a locally finite, infinite p-group and H is a finite abelian group whose order is not divisible by p, (ii) if K is a field of characteristic p > 0 and G = P × H where P is an infinite locally finite p-group (not necessarily nilpotent) and H is a finite group whose order is not divisible by p then KG is CS if and only if H is abelian. Furthermore, commutative semilocal group algebra is always continuous and for PI group algebras this holds for local group algebras; however this result is not true, in general.

Multiplicity estimates on commutative algebraic groups

Multiplicity estimates on commutative algebraic groups

On a local-global principle for the divisibility of a rational point by a positive integer

Following two previous papers (R. Dvornicich and U. Zannier, Bull. Soc. Math. France 129 (2001), 317–338; C. R. Acad. Sci. Paris, Ser. I 338 (2004) 47–50), we continue the investigation of a local-global principle for the divisibility by a positive integer of a rational point on a commutative algebraic group. In the first half of this paper some new affirmative results are obtained for elliptic curves. In the second half we investigate the structure of possible situations when the principle does not hold; it is shown that whenever a certain abstract cohomology group does not vanish (which ‘often’ happens) there exist negative examples over suitable number fields.

Units of commutative group algebra with involution

Let p be an odd prime, F the field of p elements and G a finite abelian p-group with an arbitrary involutory automorphism. Extend this auto- morphism to the group algebra FG and consider the unitary and the symmetric normalized units of FG. This paper provides bases and determines the invariants of the two subgroups formed by these units.

Isomorphism of commutative group algebras of $p$-mixed splitting groups over rings of characteristic zero

Isomorphism of commutative group algebras of $p$-mixed splitting groups over rings of characteristic zero

Cardinal invariants for commutative group algebras

Cardinal invariants for commutative group algebras

Totally projective unit groups in modular abelian group algebras

Totally projective unit groups in modular abelian group algebras