Some Basic Theorems on 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 study focused on students’ inability to correctly use psychometric tests applied for complex numbers in the context of generating an Algebraic abelian group (closure, commutative, associatively, inverse, and identity). This was done using a psychometric test (Kuder and Richardson formula) and APOS Theory after they had been taught the relevant content. The 41 participants were second-year students in the Department of Statistics at the Federal Polytechnic Ile-Oluji. They were given a class test comprising eleven (11) objective questions, and 4 questions relating to group theory were extracted for this analysis. After that, six (6) students were interviewed to gain insight into their perspectives. It was observed that most of the students had difficulty understanding the concept of algebraic groups about complex numbers, founded on not seeing the relevance in the prerequisite knowledge of an algebraic group. Also, if the reliability test of the question is unacceptable, then a good percentage of the participant lacked the mental construction relating to APOS to answer such a question. 
 Keywords: Complex Number, Algebraic group, Prerequisite, Difficulty, Discrimination index, Psychometrics.

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

In his seminal paper, half a century ago, Hyman Bass established commutator formulas for a (stable) general linear group, which were the key step in defining the group K 1. Namely, he proved that for an associative ring A with identity, $$ E(A)=\left[E(A),E(A)\right]=\left[\mathrm{GL}(A),\mathrm{GL}(A)\right], $$ where GL(A) is the stable general linear group and E(A) is its elementary subgroup. Since then, various commutator formulas have been studied in stable and non-stable settings for classical groups, algebraic groups, and their analogs, and mostly in relation to subnormal subgroups of these groups. The basic classical theorems and methods developed for their proofs are associated with the names of the heroes of classical algebraic K-theory: Bak, Quillen, Milnor, Suslin, Swan, Vaserstein, and others. One of the dominant techniques in establishing commutator type results is localization. In the present paper, some recent applications of localization methods to the study (higher/relative) commutators in the groups of points of algebraic and algebraic-like groups, such as general linear groups GL(n,A), unitary groups GU(2n,A, Λ), and Chevalley groups G(Φ,A), are described. Some auxiliary results and corollaries of the main results are also stated. The paper provides a general overview of the subject and covers the current activities. It contains complete proofs borrowed from our previous papers and expositions of several main results to give the reader a self-contained source.

Applications to cryptography of twisting commutative algebraic groups

Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.

In this thesis we study Galois representations corresponding to abelian varieties with certain reduction conditions. We show that these conditions force the image of the representations to be big, so that the Mumford-Tate conjecture (:= MT) holds. We also prove that the set of abelian varieties satisfying these conditions is dense in a corresponding moduli space. The main results of the thesis are the following two theorems. Theorem A: Let A be an absolutely simple abelian variety, End° (A) = k : imaginary quadratic field, g = dim(A). Assume either dim(A) ≤ 4, or A has bad reduction at some prime ϕ, with the dimension of the toric part of the reduction equal to 2r, and gcd(r,g) = 1, and (r,g) ≠ (15,56) or (m -1, m(m+1)/2). Then MT holds. Theorem B: Let M be the moduli space of abelian varieties with fixed polarization, level structure and a k-action. It is defined over a number field F. The subset of M(Q) corresponding to absolutely simple abelian varieties with a prescribed stable reduction at a large enough prime ϕ of F is dense in M(C) in the complex topology. In particular, the set of simple abelian varieties having bad reductions with fixed dimension of the toric parts is dense. Besides this we also established the following results: (1) MT holds for some other classes of abelian varieties with similar reduction conditions. For example, if A is an abelian variety with End° (A) = Q and the dimension of the toric part of its reduction is prime to dim( A), then MT holds. (2) MT holds for Ribet-type abelian varieties. (3) The Hodge and the Tate conjectures are equivalent for abelian 4-folds. (4) MT holds for abelian 4-folds of type II, III, IV (Theorem 5.0(2)) and some 4-folds of type I. (5) For some abelian varieties either MT or the Hodge conjecture holds.

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.

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.

TABLE OF CONTENTS INTRODUCTION. Historical background. Summary. Notation and terminology. CHAPTER I. ALGEBRAIC MATRIC GROUPS. 1. Reducibility of sets of matrices. 2. Algebraic matric groups. 3. Jordan-Holder-Schreier theorem. 4. Commutator groups. 5. Solvable algebraic matric groups. 6. Anticompact and quasicompact algebraic matric groups. 7. Reducibility to triangular form. 8. Algebraic matric groups with certain types of normal chains. CHAPTER II. SOME P.ESULTS FROM THE THEORY OF ALGEBRAIC DIFFERENTIAL EQUATIONS. 9. Differential rings, fields, and ideals. 10. Differential polynomials. 11. Solutions. 12. Relative isomorphisms. 13. Order. 14. Dependence. 15. Homogeneous linear ordinary differential equations. CHAPTER III. NORMAL DIFFERENTIAL EXTENSION FIELDS. 16. Normal differential extension fields. CHAPTER IV. PICCARD-VESSIOT EXTENSIONS. 17. Picard-Vessiot extensions and their isomorphisms. 18. Normality. 19. Characterization of G. 20. Dimension. 21. Adjunction of new elements. 22. Linear reducibility of L(y). CHAPTER V. LIOUVILLIAN EXTENSIONS. 23. Integrals and exponentials of intgrals. 24. Liouvillian extensions. 25. The principal theorem and some consequences. 26. The proof, first half. 27. The proof, second half. REFERENCES INTRODUCTION

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 .

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.

We relate the Brauer group of a Kummer surface to the Brauer group of the corresponding abelian surface. For many pairs of elliptic curves over the rational numbers we prove that the Kummer surface attached to their product has trivial Brauer group.

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.

Existence of abelian group code partitions