Commutative Algebraic Groups

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.

This chapter deals with results about commutative linear algebraic groups which are basic for the theory of the later chapters. The important tori are introduced in 3.2, and we prove the classification theorem 3.4.9 of connected one dimensional groups. The notations are as in the previous chapters.

Applications to cryptography of twisting commutative algebraic groups

Let $X$ be a projective variety over an algebraically closed field $k$ of characteristic 0.We consider categories of rational maps from $X$ to commutative algebraic groups, and ask for objects satisfying the universal mapping property.A necessary and sufficient condition for the existence of such universal objects is given, as well as their explicit construction, using duality theory of generalized 1-motives. An important application is the Albanese of a singular projective variety, which was constructed by Esnault, Srinivas and Viehweg as a universal regular quotient of a relative Chow group of 0-cycles of degree 0 modulo rational equivalence.We obtain functorial descriptions of the universal regular quotient and its dual 1-motive.

This is a survey focusing on the Hasse principle for divisibility of points in commutative algebraic groups and its relation with the Hasse principle for divisibility of elements of the Tate–Shavarevich group in the Weil–Châtelet group. The two local-global subjects arose as a generalization of some classical questions considered respectively by Hasse and Cassels. We describe the deep connection between the two problems and give an overview of the long-established results and the ones achieved during the last twenty years, when the questions were taken up again in a more general setting. In particular, by connecting various results about the two problems, we describe how some recent developments in the first of the two local-global questions imply an answer to Cassels’ question, which improves all the results published before about that problem. This answer is best possible over {mathbb Q}. We also describe some links with other similar questions, for example the Support Problem and the local-global principle for existence of isogenies of prime degree in elliptic curves.

In this work we introduce reciprocity functors, construct the associated K-group of a family of reciprocity functors, which itself is a reciprocity functor, and compute it in several different cases. It may be seen as a first attempt to get closed to the notion of reciprocity sheaves imagined by B. Kahn. Commutative algebraic groups, homotopy invariant Nisnevich sheaves with transfers, cycle modules or Kahler differentials are examples of reciprocity functors. As commutative algebraic groups do, reciprocity functors are equipped with symbols and satisfy a reciprocity law for curves.

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.

This paper analyzes some of the fundamental results of the unit groups of the commutative modular group algebras. We introduce a more visible and concise form of a number of these results. We consider with a corresponding justification some essentially inexact and unproved results in some papers and for certain of them either we mark the corrections or we point out the corrections, which are founded in another papers.

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.

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.

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference closed field (?, σ) the ?-primary torsion of G(?) is contained in a modular group definable in (?, σ), then it is contained in a group of the form {x∈G(?) :σ(x) =[a](x) } with a∈ℕ\pℕ. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall.

A cyclic code is an ideal in the group algebra of a special kind of Abelian group, namely a cyclic group. Many properties of cyclic codes are special cases of properties of ideals in an Abelian group algebra. A character of an Abelian group G of order v is, for our purposes, a homomorphism of G into the group of vth roots of unity over GF(2). If G is cyclic with generator x, the character is entirely determined by what it does to x; this effect is kept, and the characters are discarded. If G is not cyclic it is necessary to rehabilitate the characters. Without them the notation is impossible; with them one can prove a number of theorems which reduce in the special case to well-known properties of cyclic codes. Moreover the writer thinks that the general proof is often easier and more suggestive than the proof for the special case. To support this point of view we produce a new theorem, which of course also applies to cyclic codes.

Consider the abelian category \mathcal{C}_k of commutative group schemes of finite type over a field k . By results of Serre and Oort, \mathcal{C}_k has homological dimension 1 (resp. 2) if k is algebraically closed of characteristic 0 (resp. positive). In this article, we explore the abelian category of commutative algebraic groups up to isogeny, defined as the quotient of \mathcal{C}_k by the full subcategory \mathcal{F}_k of finite k -group schemes. We show that \mathcal{C}_k/\mathcal{F}_k has homological dimension 1, and we determine its projective or injective objects. We also obtain structure results for \mathcal{C}_k/\mathcal{F}_k , which take a simpler form in positive characteristics.

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.

Characteristic principal bundles are the duals of commutative twisted group algebras. A principal bundle with locally compact second countable (Abelian) group and base space is characteristic iff it supports a continuous eigenfunction for almost every character measurably in the characters, also iff it is the quotient by Z of a principal E-bundle for every E in Ext ( G , Z ) {\operatorname {Ext}}(G,Z) and a measurability condition holds. If a bundle is locally trivial, n.a.s.c. for it to be such a quotient are given in terms of the local transformations and Čech cohomology of the base space. Although characteristic G-bundles need not be locally trivial, the class of characteristic G-bundles is a homotopy invariant of the base space. The isomorphism classes of commutative twisted group algebras over G with values in a given commutative C ∗ {C^\ast } -algebra A are classified by the extensions of G by the integer first Čech cohomology group of the maximal ideal space of A.