Commutative Algebraic Groups Research Articles

Transcendence on algebraic groups

Abstract In this paper, we give some new results on transcendence on algebraic groups. These results extend some previous ones established on commutative or linear algebraic groups to arbitrary algebraic groups in complex and p-adic fields, respectively.

Completeness of derivation algebras of finite commutative group algebras

Completeness of derivation algebras of finite commutative group algebras

Unit groups of commutative modular group algebras

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 other articles.

Torsion of algebraic groups and iterate extensions associated with Lubin–Tate formal groups

We show finiteness results on torsion points of commutative algebraic groups over a $p$-adic field $K$ with values in various algebraic extensions $L/K$ of infinite degree. We mainly study the following cases: (1) $L$ is an abelian extension which is a splitting field of a crystalline character (such as a Lubin–Tate extension). (2) $L$ is a certain iterate extension of $K$ associated with points on Lubin–Tate formal groups.

Burst error-correcting quantum stabilizer codes designed from idempotents

Certain classical codes can be viewed isomorphically as ideals of group algebras, while studying their algebraic structures help extracting the code properties. Research has shown that this was remarkably efficient in the case when the code generators are idempotents. In quantum error correction, the theory of stabilizer formalism requires classical self-orthogonal additive codes over the finite field GF(4), which, via the lens of group algebras, are essentially F_2-submodules over GF(4). Therefore, this paper provides a classification on idempotents in commutative group algebra GF(4)G, followed by a criterion that allows idempotents to generate stabilizer subgroups. Later, the construction of quantum stabilizer codes is done in the case when G is a cyclic group C_n, for n=2^m-1 and n=2^m+1. Quantum bounds on their burst error minimum distance are subsequently determined.

Strongly regular matrices revisited

We prove a necessary condition and a sufficient condition for an n×n-matrix A with determinantal rank ρ(A)=t over an arbitrary commutative ring to be (von Neumann) strongly regular in terms of the trace of its tth compound matrix Ct(A). In particular, a non-zero n×n-matrix A with ρ(A)=t over a local commutative ring R is strongly regular if and only if Tr(Ct(A)) is a unit in R, and in this case we construct a strong inner inverse of A. We derive applications to products of local commutative rings and group algebras. Finally, we count strongly regular matrices over some finite rings of residue classes and group algebras.

Integral Models and Torsors of Inseparable Forms of Ga

After recalling some basic facts about F-wound commutative unipotent algebraic groups over an imperfect field F, we study their regular integral models over Dedekind schemes of positive characteristic and compute the group of isomorphism classes of torsors of one-dimensional groups.

On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0

AbstractLet K be a field of characteristic 0 and let G and H be connected commutative algebraic groups over K. Let Mor0(G,H) denote the set of morphisms of algebraic varieties G → H that map the neutral element to the neutral element. We construct a natural retraction from Mor0(G,H) to Hom(G,H) (for arbitrary G and H) which commutes with the composition and addition of morphisms. In particular, if G and H are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If G has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between G and H. We also characterize all connected commutative algebraic groups over K whose only variety automorphisms are compositions of automorphisms of algebraic groups with translations.

Degree one Milnor K-invariants of groups of multiplicative type

Let G be a commutative affine algebraic group over a field F, and let H:FieldsF→AbGrps be a functor. A (homomorphic) H-invariant of G is a natural transformation Tors(−,G)→H, where Tors(−,G) is the functor FieldsF→AbGrps taking a field extension L/F to the group of isomorphism classes of GL-torsors over Spec(L). The goal of this paper is to compute the group Invhom1(G,H) of H-invariants of G when G is a group of multiplicative type, and H is the functor taking a field extension L/F to L×⊗ZQ/Z.

Cohomology of bimultiplicative local systems on unipotent groups

Let U1,U2 be connected commutative unipotent algebraic groups defined over an algebraically closed field k of characteristic p>0 and let L be a bimultiplicative Q‾ℓ-local system on U1×U2. In this paper we will study the Q‾ℓ-cohomology Hc⁎(U1×U2,L), which turns out to be supported in only one degree. We will construct a finite Heisenberg group Γ which naturally acts on Hc⁎(U1×U2,L) as an irreducible representation. We will give two explicit realizations of this cohomology and describe the relationship between these two realizations as a finite Fourier transform.

Projective geometries arising from Elekes-Szabó problems

We generalise the Elekes-Szab\'o theorem to arbitrary arity and dimension and characterise the complex algebraic varieties without power saving. The characterisation involves certain algebraic subgroups of commutative algebraic groups endowed with an extra structure arising from a skew field of endomorphisms. We also extend the Erd\H{o}s-Szemer\'edi sum-product phenomenon to elliptic curves. Our approach is based on Hrushovski's framework of pseudo-finite dimensions and the abelian group configuration theorem.

Unitary Subgroups of Commutative Group Algebras of the Characteristic Two

Let FG be the group algebra of a finite 2-group G over a finite field F of characteristic two and let ⊛ be an involution that arises from G. The ⊛-unitary subgroup of FG is denoted by V⊛(FG) and defined as the set of all normalized units u satisfying the property u⊛ = u−1. We establish the order of V⊛(FG) for all involutions ⊛ arising from G, where G is a finite cyclic 2-group, and show that all ⊛-unitary subgroups of FG are not isomorphic.

On the moments of torsion points modulo primes and their applications

Let [Formula: see text] be the group of [Formula: see text]-torsion points of a commutative algebraic group [Formula: see text] defined over a number field [Formula: see text]. For a prime [Formula: see text] of [Formula: see text], we let [Formula: see text] be the number of [Formula: see text]-solutions of the system of polynomial equations defining [Formula: see text] when reduced modulo [Formula: see text]. Here, [Formula: see text] is the residue field at [Formula: see text]. Let [Formula: see text] denote the number of primes [Formula: see text] of [Formula: see text] such that [Formula: see text]. We then, for algebraic groups of dimension one, compute the [Formula: see text]th moment limit [Formula: see text] by appealing to the Chebotarev density theorem. We further interpret this limit as the number of orbits of the action of the absolute Galois group of [Formula: see text] on [Formula: see text] copies of [Formula: see text] by an application of Burnside’s Lemma. These concrete examples suggest a possible approach for determining the number of orbits of a group acting on [Formula: see text] copies of a set.

On the topology of geometric and rational orbits for algebraic group actions over valued fields

Abstract In this note, we study the relationship between Zariski and relative closedness for actions of (smooth) algebraic groups defined over valued (mainly local) fields of any characteristic. In particular, we use some recent basic results regarding the completely reducible subgroups and cocharacter-closedness due to Bate–Herpel–Röhrle–Tange and Uchiyama to construct some actions of simple algebraic groups G of the types D 4 {D_{4}} , E 6 {E_{6}} , E 7 {E_{7}} , E 8 {E_{8}} , G 2 {G_{2}} on an affine variety defined over a local function field k, and v ∈ V ⁢ ( k ) {v\in V(k)} such that the geometric orbit G . v {G.v} is Zariski closed although the corresponding relative orbit G ⁢ ( k ) . v {G(k).v} is not closed in the topology induced from k. Besides, by using an interesting result due to Gabber, Gille and Moret-Bailly, we show that this phenomenon does not appear when we consider the action of either a smooth unipotent group or a smooth commutative algebraic group, defined over an admissible valued (e.g., local) field.

Translation-invariant line bundles on linear algebraic groups

We study the Picard groups of connected linear algebraic groups and especially the subgroup of translation-invariant line bundles. We prove that this subgroup is finite over every global function field. We also utilize our study of these groups in order to construct various examples of pathological behavior for the cohomology of commutative linear algebraic groups over local and global function fields.

Free pairs of symmetric elements in normal subgroups of division rings

Let D be a division ring with central subfield k of characteristic ≠2, let ⁎ be a k-involution of D, and let N⊲D• be a normal subgroup of the multiplicative group D• of D. We show that if G is a ⁎-stable nonabelian subgroup of N that is either torsion-free polycyclic-by-finite but not abelian-by-finite, or finite of odd order, then N contains a pair (u,v) of elements such that u⁎=u, v⁎=v and such that 〈u,v〉 is a noncyclic free group.One aspect of the above proof requires that we extend a theorem of Bergman on invariant ideals in commutative group algebras. This new result is surely of interest in its own right. It appears in the Appendix and can be read independently of the remainder of the paper.

On the fundamental groups of commutative algebraic groups

Consider the abelian category ${\mathcal C}$ of commutative group schemes of finite type over a field $k$, its full subcategory ${\mathcal F}$ of finite group schemes, and the associated pro category ${\rm Pro}({\mathcal C})$ (resp. ${\rm Pro}({\mathcal F})$) of pro-algebraic (resp. profinite) group schemes. When $k$ is perfect, we show that the profinite fundamental group $\varpi_1 : {\rm Pro}({\mathcal C}) \to {\rm Pro}({\mathcal F})$ is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors $\varpi_i$ vanish for $i \geq 2$. Along the way, we describe the indecomposable projective objects of ${\rm Pro}({\mathcal C})$ over an arbitrary field $k$.

Divisibility questions in commutative algebraic groups

Let k be a number field, let A be a commutative algebraic group defined over k and let p be a prime number. Let A[p] denote the p-torsion subgroup of A. We give some sufficient conditions for the local-global divisibility by p in A and the triviality of the Tate-Shafarevich group ▪. When A is a principally polarized abelian variety, those conditions imply that the elements of the Tate-Shafarevich group ▪ are divisible by p in the Weil-Châtelet group H1(k,A) and the local-global principle for divisibility by p holds in Hr(k,A), for all r≥0.

Some remarks on K2 and K3 of finite abelian group algebras

Let [Formula: see text] be a finite abelian [Formula: see text]-group and [Formula: see text] a finite field of characteristic [Formula: see text]. We give an algorithm to write down a set of generators of [Formula: see text] via a simple presentation of it and obtain some formulae for counting these generators. Using Witt decomposition, we also determine the group structure of [Formula: see text].

F-sets and finite automata

It is observed that Derksen’s Skolem–Mahler–Lech theorem is a special case of the isotrivial positive characteristic Mordell-Lang theorem due to the second author and Scanlon. This motivates an extension of the classical notion of a k-automatic subset of the natural numbers to that of an F-automatic subset of a finitely generated abelian group Γ equipped with an endomorphism F. Applied to the Mordell–Lang context, where F is the Frobenius action on a commutative algebraic group G over a finite field, and Γ is a finitely generated F-invariant subgroup of G, it is shown that the “F-subsets” of Γ introduced by the second author and Scanlon are F-automatic. It follows that when G is semiabelian and X⊆G is a closed subvariety then X∩Γ is F-automatic. Derksen’s notion of a k-normal subset of the natural numbers is also here extended to the above abstract setting, and it is shown that F-subsets are F-normal. In particular, the X∩Γ appearing in the Mordell-Lang problem are F-normal. This generalises Derksen’s Skolem–Mahler–Lech theorem to the Mordell–Lang context.