Deciding finiteness for matrix semigroups over function fields over finite fields

We present a deterministic polynomial time algorithm for testing finiteness of a semigroupS generated by matrices with entries from function fields of constant transcendence degree over finite fields. A special case of the problem was shown to be algorithmically soluble in [RTB] by giving a sharp exponential upper bound on the dimension of the matrix algebra generated byS over the field of constants. One of the exponential time algorithms proposed in [RTB] was expected to be improvable. The polynomial time method presented in this note combines the ideas of that algorithm with a procedure from [IRSz] for calculating the radical. Research supported by NWO-OTKA Grant N26673, FKFP Grant 0612/1997, OTKA Grants 016503, 022925, and EC Grant ALTEC-KIT.     

Deciding finiteness of matrix groups in positive characteristic

We present a new algorithm to decide finiteness of matrix groups defined over a field of positive characteristic. Together with previous work for groups in zero characteristic, this provides the first complete solution of the finiteness problem for finitely generated matrix groups over a field. We also give an algorithm to compute the order of a finite matrix group over a function field of positive characteristic by constructing an isomorphic copy of the group over a finite field. Our implementations of these algorithms are publicly available in Magma.     

A local-global principle for finiteness properties ofS-arithmetic groups over number fields

In the preceding paper [AT] compactness propertiesC _n andCP _n for locally compact groups were introduced. They generalize the finiteness propertiesF _n andFP _n for discrete groups. In this paper a local-global principle forS-arithmetic groups over number fields is proved. TheS-arithmetic group is of typeF _n , resp.FP _n , if and only if for allp inS thep-adic completionG _p of the corresponding algebraic groupG is of typeC _n resp.CP _n . As a corollary we obtain an easy proof of a theorem of Borel and Serre: AnS-arithmetic subgroup of a semisimple group has all the finiteness propertiesF _n .     

Hasse principle for classical groups over function fields of curves over number fields

In (Letter to J.-P. Serre, 12 June 1991) Colliot-Thélène conjectures the following: Let F be a function field in one variable over a number field, with field of constants k and G be a semisimple simply connected linear algebraic group defined over F. Then the map has trivial kernel, denoting the set of places of k.The conjecture is true if G is of type 1A∗, i.e., isomorphic to SL₁(A) for a central simple algebra A over F of square free index, as pointed out by Colliot-Thélène, being an immediate consequence of the theorems of Merkurjev-Suslin [S1] and Kato [K]. Gille [G] proves the conjecture if G is defined over k and F=k(t), the rational function field in one variable over k. We prove that the conjecture is true for groups G defined over k of the types 2A∗, B_n, C_n, D_n (D₄ nontrialitarian), G₂ or F₄; a group is said to be of type 2A∗, if it is isomorphic to SU(B,τ) for a central simple algebra B of square free index over a quadratic extension k′ of k with a unitary k′|k involution τ.     

Pro-P galois groups of function fields over local fields

For non-archimedean local field K and a prime number p we compute the finitely generated pro-p (closed) subgroups of the absolute Galois group of K(t). In addition, we characterize the finitely generated pro-p groups which occur as the maximal pro-p Galois group of algebraic extensions of K(t) containing a primitive pth root of unity.     

Finiteness properties of arithmetic groups over function fields

We determine when an arithmetic subgroup of a reductive group defined over a global function field is of type FP _∞ by comparing its large-scale geometry to the large-scale geometry of lattices in real semisimple Lie groups.     

Demjanenko matrix and recursion formula for relative class number over function fields

In this paper, we define Demjanenko matrix in function field and express the relative ideal class numbers as the determinant of this matrix. We also define another matrix which give us a recursion formula for the relative divisor class numbers h⁻(K_Pⁿ).     

Connected components of compact matrix quantum groups and finiteness conditions

We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of constructing the identity component by introducing canonical approximating transfinite sequences of subgroups. These sequences have lengths ≤1 in the classical case but can be countably infinite for duals of discrete groups. We give examples of free product quantum groups where the identity component is not normal and the associated sequence has length 1.     

On Euler characteristics of Selmer groups for abelian varieties over global function fields

Let F be a global function field of characteristic \({p > 0}\), \({K/F}\) an \({\ell}\)-adic Lie extension (\({ \ell \neq p}\)), and \({A/F}\) an abelian variety. We provide Euler characteristic formulas for the Gal\({(K/F)}\)-module \({Sel_A(K)_\ell}\).     

Conditions for finiteness in Aleshin-type groups

Translated from Algebra i Logika, Vol. 29, No. 6, pp. 724–745, November–December, 1990.     

A hasse principle for function fields over pac fields

LetK be a perfect pseudo-algebraically closed field and letF be an extension ofK of relative transcendence degree 1. It is shown that the restriction map Res: Br(F)→Π_p Br(F _p ^h ) is injective, where p ranges over all non-trivialK-places ofF, andF _p ^h is the corresponding henselization. Conversely, the validity of this Hasse principle for all such extensionsF implies a weaker version of pseudo-algebraic closedness. As an application we determine the finitely generated pro-p closed subgroups of the absolute Galois group ofK(t).     

Equidistribution over function fields

We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For number fields, these results were proved by Yuan and we transfer here his methods to function fields. If X is a closed subvariety of an abelian variety, then we can describe the equidistribution measure explicitly in terms of convex geometry.     

Galois groups over complete valued fields

We propose an elementary algebraic approach to the patching of Galois groups. We prove that every finite group is regularly realizable over the field of rational functions in one variable over a complete discrete valued field. Partially supported by NSF grant DMS 9306479.     

Frobenius Galois groups over quadratic fields

There exists a quadratic fieldQ(√D) over which every Frobenius group is realizable as a Galois group.     

Contractible Lie groups over local fields

Let G be a Lie group over a local field of characteristic p > 0 which admits a contractive automorphism α : G → G (i.e., α ⁿ (x) → 1 as n → ∞, for each x ∈ G). We show that G is a torsion group of finite exponent and nilpotent. We also obtain results concerning the interplay between contractive automorphisms of Lie groups over local fields, contractive automorphisms of their Lie algebras, and positive gradations thereon. Some of the results extend to Lie groups over arbitrary complete ultrametric fields. Supported by the German Research Foundation (DFG), grants GL 357/2-1 and GL 357/6-1.