The arithmetic theory of algebraic groups

In this paper we prove local–global principles for the existence of an embedding (E, σ) ↪ (A, τ) of a given global field E endowed with an involutive automorphism σ into a simple algebra A given with an involution τ in all situations except where A is a matrix algebra of even degree over a quaternion division algebra and τ is orthogonal (Theorem A of the introduction). Rather surprisingly, in the latter case we have a result which in some sense is opposite to the local–global principle, viz. algebras with involution locally isomorphic to (A, τ) are distinguished by their maximal subfields invariant under the involution (Theorem B of the introduction). These results can be used in the study of classical groups over global fields. In particular, we use Theorem B to complete the analysis of weakly commensurable Zariski-dense S -arithmetic groups in all absolutely simple algebraic groups of type different from D_4 which was initiated in our paper [23]. More precisely, we prove that in a group of type D_n , n even > 4 , two weakly commensurable Zariski-dense S -arithmetic subgroups are actually commensurable. As indicated in [23], this fact leads to results about length-commensurable and isospectral compact arithmetic hyperbolic manifolds of dimension 4n + 7 , with n ≥ 1 . The appendix contains a Galois-cohomological interpretation of our embedding theorems.

A subexponential algorithm for discrete logarithms over hyperelliptic curves of large genus over GF( q)

In this chapter we will prove Mordell’s theorem that the group of rational points on a non-singular cubic is finitely generated. There is a tool used in the proof called the height. In brief, the height of a rational point measures how complicated the point is from the viewpoint of number theory.

In [1], Borel discussed discrete arithmetic groups arising from quaternion algebras over number fields with particular reference to arithmetic Kleinian and arithmetic Fuchsian groups. In these cases, he described, in each commensurability class, a class of groups which contains all maximal groups. Developing results on embedding commutative orders of the defining number field into maximal or Eichler orders in the defining quaternion algebra, Chinburg and Friedman [2] stated necessary and sufficient conditions for the existence of torsion in this class of groups in terms of the defining arithmetic data. This was more fully explored in the case of Kleinian groups in [3]. In the case of Fuchsian groups, these results on the existence of torsion were extended to obtain formulas for the number of conjugacy classes of finite cyclic subgroups for each group in this class [8, 9]. In this paper, we examine, across the range of arithmetic Fuchsian groups, how widespread torsion is in maximal Fuchsian groups. Some studies in low genus cases (see e.g. [7, 12]) indicate that 2-torsion is very prevalent. The results obtained here substantiate that but we will also obtain maximal arithmetic Fuchsian groups which are torsion-free. The author is grateful to Alan Reid for conversations on parts of this paper.

There are errors in the proof of uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture, and we give several remarks on further developments. Mathematics Subject Classification (2000): 11E57 (primary); 22E40 (secondary). 1.1. Let us recall some notation and basic notions. Following [1] we will assume that n is even and n ≥ 4. The group of orientation preserving isometries of hyperbolic n-space is isomorphic to SO(1, n)o, the connected component of the identity of the special orthogonal group of signature (1, n), which can be identified with SO0(1, n), the subgroup of SO(1, n) preserving the upper half space. This group is not Zariski closed in SLn+1 thus in order to construct arithmetically defined subgroups of SO(1, n)o we consider arithmetic subgroups of the orthogonal group SO(1, n) or, more precisely, of groups G = SO( f ) where f is an admissible quadratic form defined over a totally real number field k (see [1, Section 2.1]). We have an exact sequence of k-isogenies: 1 → C → G φ → G → 1, (1.1) where G(k) Spin( f ) is the simply connected cover of G and C μ2 is the center of G. This induces an exact sequence in Galois cohomology (see [5, Section 2.2.3]) G(k) φ → G(k) δ → H1(k, C) → H1(k, G). (1.2) The main idea of this note is that by using (1.2) certain questions about arithmetic subgroups of G can be reduced to questions about the Galois cohomology group H1(k, C). A coherent collection of parahoric subgroups P = (Pv)v∈V f of G (V f = V f (k) denotes the set of finite places of the field k) defines a principal arithmetic subgroup Received October 16, 2006; accepted in revised form March 12, 2007.

Let G be a connected linear algebraic group over a number field k. We establish an exact sequence describing the closure of the group G(k) of rational points of G in the group of adelic points of G. This exact sequence describes the defect of strong approximation on G in terms of the algebraic Brauer group of G. In particular, we deduce from those results that the integral Brauer-Manin obstruction on a torsor under the group G is the only obstruction to the existence of an integral point on this torsor. We also obtain a non-abelian Poitou-Tate exact sequence for the Galois cohomology of the linear group G. The main ingredients in the proof of those results are the local and global duality theorems for complexes of k-tori of length two and the abelianization maps in Galois cohomology introduced by Borovoi.

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.

An elliptic curve can be defined as a smooth projective curve of degree 3 in the projective plane with a distinguished point chosen on it. The set of points on the curve can thus be endowed with a natural additive group structure. The most concrete description of an elliptic curve comes from its affine equation, written as $$y^2=x^3+ax+b,\quad \text{with}\ 4a^3+27b^2\not=0.$$ The theory of elliptic curves is a marvelous mixture of elementary mathematics and profound, advanced mathematics, a mixture which moreover lies on the crossroads of multiple theories: arithmetic, algebraic geometry, group representations, complex analysis, etc. Here, we will provide an introduction to the subject and prove the main Diophantine theorems: the group of rational points is finitely generated (the Mordell-Weil theorem) and the set of integral points is finite (Siegel’s theorem). Finally, we will evoke the famous theorem of Wiles—whose proof resulted in the proof of Fermat’s last theorem—and the Birch & Swinnerton-Dyer conjecture.

The constructive theory of hyperelliptic curves has been advanced significantly during the last year. It is intended to give here an indication of the current level of progress, and an outline of the main methods employed. The emphasis in Sections 1 to 4 will be on the group of rational points on the Jacobian of a hyperelliptic curve. Section 5 will concern itself with the use of the Jacobian to help to determine the rational points on the curve itself.KeywordsAbelian GroupElliptic CurveRational PointElliptic CurfNumber FieldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we construct an authentication scheme for multi-receivers and multiple messages based on a linear code C. This construction can be regarded as a generalization of the authentication scheme given by Safavi-Naini and Wang [1]. Actually, we notice that the scheme of Safavi-Naini and Wang is constructed with Reed-Solomon codes. The generalization to linear codes has the similar advantages as generalizing Shamir's secret sharing scheme to linear secret sharing sceme based on linear codes [2—6]. For a fixed message base field ${\mathbb F}_q$ , our scheme allows arbitrarily many receivers to check the integrity of their own messages, while the scheme of Safavi-Naini and Wang has a constraint on the number of verifying receivers $V\leqslant q$ . We further introduce access structure in our scheme. Masseya[4] characterized the access structure of linear secret sharing scheme by minimal codewords in the dual code whose first component is 1. We slightly modify the definition of minimal codewords in [4]. Let C be a [V,k] linear code. For any coordinate i ∈ {1,2, i¾? ,V}, a codeword c in C is called minimal respect to i if the codeword c has component 1 at the i-th coordinate and there is no other codeword whose i-th component is 1 with support strictly contained in that of c. Then the security of receiver R i in our authentication scheme is characterized by the minimal codewords respect to i in the dual code $C^\bot$ . Finally, we illustrate our authentication scheme based on the elliptic curve codes, a special class of algebraic geometry codes. We use the group of rational points on the elliptic curve to determine all the malicious groups that can successfully make a substitution attack to any fixed receiver.

An isogeny class $\mathcal{A}$ of abelian varieties defined over finite fields is said to be "cyclic" if every variety in $\mathcal{A}$ has a cyclic group of rational points. In this paper we study the local cyclicity of Weil-central isogeny classes of abelian varieties, i.e. those with Weil polynomials of the form $f_\mathcal{A}(t)=t^{2g}+at^g+q^g$, as well as the local growth of the groups of rational points of the varieties in $\mathcal{A}$ after finite field extensions. We exploit the criterion: an isogeny class $\mathcal{A}$ with Weil polynomial $f$ is cyclic if and only if $f'(1)$ is coprime with $f(1)$ divided by its radical.

The group SL(3,Z) cannot act (faithfully) on the circle (by homeomorphisms). We will see that many other arithmetic groups also cannot act on the circle. The discussion will involve several important topics in group theory, such as ordered groups, amenability, bounded generation, and bounded cohomology. Lecture 1 provides an introduction to the subject, and uses the theory of left-orderable groups to prove that SL(3,Z) does not act on the circle. Lecture 2 discusses bounded generation, and proves that groups of the form SL(2,Z[a]) do not act on the real line. Lectures 3 and 4 are brief introductions to amenable groups and bounded cohomology, respectively. They also explain how these ideas can be used to prove that actions on the circle have finite orbits. An appendix provides hints or references for all of the exercises. These notes are slightly expanded from talks given at the Park City Mathematics Institute's Graduate Summer School in July 2012.

Cubic Forms - Algebra, Geometry, Arithmetic

Let X X be a smooth curve over a finite field of characteristic p p , let E E be a number field, and let L = { L λ } \mathbf {L} = \{\mathcal {L}_\lambda \} be an E E -compatible system of lisse sheaves on the curve X X . For each place λ \lambda of E E not lying over p p , the λ \lambda -component of the system L \mathbf {L} is a lisse E λ E_\lambda -sheaf L λ \mathcal {L}_\lambda on X X , whose associated arithmetic monodromy group is an algebraic group over the local field E λ E_\lambda . We use Serre’s theory of Frobenius tori and Lafforgue’s proof of Deligne’s conjecture to show that when the E E -compatible system L \mathbf {L} is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is “independent of λ \lambda ”. More precisely, after replacing E E by a finite extension, there exists a connected split reductive algebraic group G 0 G_0 over the number field E E such that for every place λ \lambda of E E not lying over p p , the identity component of the arithmetic monodromy group of L λ \mathcal {L}_\lambda is isomorphic to the group G 0 G_0 with coefficients extended to the local field E λ E_\lambda .

We show that the abstract commensurator of an S-arithmetic subgroup of a solvable algebraic group over Q is isomorphic to the Q-points of an algebraic group, and compare this with examples of nonlinear abstract commensurators of S-arithmetic groups in positive characteristic. In particular, we include a description of the abstract commensurator of the lamplighter group.