scholarly journals Hilbert’s 14th problem over finite fields and a conjecture on the cone of curves

2008 ◽  
Vol 144 (5) ◽  
pp. 1176-1198 ◽  
Author(s):  
Burt Totaro
Keyword(s):  
Finite Fields ◽  
Additive Group ◽  
Rational Numbers ◽  
Abelian Surface ◽  
Finitely Generated ◽  
Finite Generation ◽  
Elliptic Fibrations ◽  
Anticanonical Bundle ◽  
Cone Of Curves

AbstractWe give the first examples over finite fields of rings of invariants that are not finitely generated. (The examples work over arbitrary fields, for example the rational numbers.) The group involved can be as small as three copies of the additive group. The failure of finite generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell–Weil rank. Our work suggests a generalization of the Morrison–Kawamata cone conjecture on Calabi–Yau fiber spaces to klt Calabi–Yau pairs. We prove the conjecture in dimension two under the assumption that the anticanonical bundle is semi-ample.

scholarly journals On a problem on explicit embeddings of the groupℚ

2005 ◽  
Vol 2005 (13) ◽  
pp. 2119-2123 ◽  
Author(s):  
Vahagn H. Mikaelian
Keyword(s):  
Additive Group ◽  
Rational Numbers ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Kourovka Notebook ◽  
Torsion Freeness ◽  
As Solubility

Answering a question of de la Harpe and Bridson in the Kourovka Notebook, we build the explicit embeddings of the additive group of rational numbersℚin a finitely generated groupG. The groupGin fact is two-generator, and the constructed embedding can be subnormal and preserve a few properties such as solubility or torsion freeness.

scholarly journals Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups

2015 ◽  
Vol 58 (4) ◽  
pp. 787-798 ◽  
Author(s):  
Yu Kitabeppu ◽  
Sajjad Lakzian
Keyword(s):  
Diameter Growth ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Finite Generation ◽  
Alexandrov Spaces ◽  
Geodesic Spaces ◽  
Special Case ◽  
Linear Diameter

AbstractIn this paper, we generalize the finite generation result of Sormani to non-branching RCD(0, N) geodesic spaces (and in particular, Alexandrov spaces) with full supportmeasures. This is a special case of the Milnor’s Conjecture for complete non-compact RCD(0, N) spaces. One of the key tools we use is the Abresch–Gromoll type excess estimates for non-smooth spaces obtained by Gigli–Mosconi.

Local Bounds for Torsion Points on Abelian Varieties

2008 ◽  
Vol 60 (3) ◽  
pp. 532-555 ◽  
Author(s):  
Pete L. Clark ◽  
Xavier Xarles
Keyword(s):  
Abelian Varieties ◽  
Residue Field ◽  
Rational Numbers ◽  
Abelian Surface ◽  
Torsion Subgroup ◽  
Good Reduction ◽  
Torsion Points ◽  
Ramification Index ◽  
Numerical Invariants ◽  
Split Torus

AbstractWe say that an abelian variety over a p-adic field K has anisotropic reduction (AR) if the special fiber of its Néronminimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the K-rational torsion subgroup of a g-dimensional AR variety depending only on g and the numerical invariants of K (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of g, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an AR abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.

scholarly journals Generators and relations of Rees matrix semigroups

1999 ◽  
Vol 42 (3) ◽  
pp. 481-495 ◽  
Author(s):  
H. Ayik ◽  
N. Ruškuc
Keyword(s):  
Rees Matrix Semigroup ◽  
Finitely Generated ◽  
Finite Generation ◽  
Generators And Relations ◽  
Rees Matrix Semigroups ◽  
Finite Presentability ◽  
Matrix Semigroup ◽  
Finitely Presented ◽  
Matrix Semigroups ◽  
The Ideal

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P.

MULTIPLICATIVE ORDERS IN ORBITS OF POLYNOMIALS OVER FINITE FIELDS

2017 ◽  
Vol 60 (2) ◽  
pp. 487-493 ◽  
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Fields ◽  
Algebraic Closure ◽  
Rational Numbers ◽  
Duke Math ◽  
Roots Of Unity ◽  
Multiplicative Order ◽  
Polynomials Over Finite Fields ◽  
Polynomial Orbits

AbstractWe show, under some natural restrictions, that orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime p. We also show that for all but finitely many initial points either the multiplicative order of this point or the length of the orbit it generates (both modulo a large prime p) is large. The approach is based on the results of Dvornicich and Zannier (Duke Math. J.139 (2007), 527–554) and Ostafe (2017) on roots of unity in polynomial orbits over the algebraic closure of the field of rational numbers.

Decision problems concerning S-arithmetic groups

1985 ◽  
Vol 50 (3) ◽  
pp. 743-772 ◽  
Author(s):  
Fritz Grunewald ◽  
Daniel Segal
Keyword(s):  
Additive Group ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Algorithmic Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Finite Dimensional ◽  
Associative Rings ◽  
Finitely Presented ◽  
Torsion Free

This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work.We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”.(1) Isomorphism problems. Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases: = {finitely generated nilpotent groups}; = {(not necessarily associative) rings whose additive group is finitely generated}; = {finitely Z-generated modules over a fixed finitely generated ring}.Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q-algebras.

scholarly journals Explicit Kummer theory for the rational numbers

2020 ◽  
Vol 16 (10) ◽  
pp. 2213-2231
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba ◽  
Sebastiano Tronto
Keyword(s):  
Rational Numbers ◽  
Finitely Generated ◽  
Finite Case ◽  
Kummer Theory ◽  
Multiplicative Subgroup

Let [Formula: see text] be a finitely generated multiplicative subgroup of [Formula: see text] having rank [Formula: see text]. The ratio between [Formula: see text] and the Kummer degree [Formula: see text], where [Formula: see text] divides [Formula: see text], is bounded independently of [Formula: see text] and [Formula: see text]. We prove that there exist integers [Formula: see text] such that the above ratio depends only on [Formula: see text], [Formula: see text], and [Formula: see text]. Our results are very explicit and they yield an algorithm that provides formulas for all the above Kummer degrees (the formulas involve a finite case distinction).

Finite presentability of generalized Bruck–Reilly ∗-extension of groups

2016 ◽  
Vol 09 (04) ◽  
pp. 1650090 ◽  
Author(s):  
Seda Oğuz ◽  
Eylem G. Karpuz
Keyword(s):  
Finite Subset ◽  
Identity Element ◽  
Finitely Generated ◽  
Finite Generation ◽  
Finite Set ◽  
Finite Presentability ◽  
Finitely Presented

In [Finite presentability of Bruck–Reilly extensions of groups, J. Algebra 242 (2001) 20–30], Araujo and Ruškuc studied finite generation and finite presentability of Bruck–Reilly extension of a group. In this paper, we aim to generalize some results given in that paper to generalized Bruck–Reilly ∗-extension of a group. In this way, we determine necessary and sufficent conditions for generalized Bruck–Reilly ∗-extension of a group, [Formula: see text], to be finitely generated and finitely presented. Let [Formula: see text] be a group, [Formula: see text] be morphisms and [Formula: see text] ([Formula: see text] and [Formula: see text] are the [Formula: see text]- and [Formula: see text]-classes, respectively, contains the identity element [Formula: see text] of [Formula: see text]). We prove that [Formula: see text] is finitely generated if and only if there exists a finite subset [Formula: see text] such that [Formula: see text] is generated by [Formula: see text]. We also prove that [Formula: see text] is finitely presented if and only if [Formula: see text] is presented by [Formula: see text], where [Formula: see text] is a finite set and [Formula: see text] [Formula: see text] for some finite set of relations [Formula: see text].

Abelian Groups with a Vanishing Homology Group

1969 ◽  
Vol 21 ◽  
pp. 406-409 ◽  
Author(s):  
James A. Schafer
Keyword(s):  
Homology Group ◽  
Abelian Groups ◽  
Additive Group ◽  
Rational Numbers ◽  
Complete Characterization ◽  
Integral Homology ◽  
Positive Dimension ◽  
Homology Groups

In this paper, we wish to characterize those abelian groups whose integral homology groups vanish in some positive dimension. We obtain a complete characterization provided the dimension in which the homology vanishes is odd; in fact, we prove that the only abelian groups which possess a vanishing homology group in an odd dimension are, up to isomorphism, subgroups of Qn, where Q denotes the additive group of rational numbers. The case of vanishing in an even dimension is much more complicated. We exhibit a class of groups whose homology vanishes in even dimensions and is otherwise very nice, namely the subgroups of Q/Z, and then show that unless we impose further restrictions, there exist abelian groups which possess the homology of subgroups of Q/Z without being isomorphic to a subgroup of Q/Z.

The Augmentation Terminals of Certain Locally Finite Groups

1972 ◽  
Vol 24 (2) ◽  
pp. 221-238 ◽  
Author(s):  
K. W. Gruenberg ◽  
J. E. Roseblade
Keyword(s):  
Homomorphic Image ◽  
Group Ring ◽  
Finite Groups ◽  
Prime Order ◽  
Additive Group ◽  
Rational Numbers ◽  
Group Element ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Locally Finite

Let G be a group and ZG be the integral group ring of G. We shall write 𝔤 for the augmentation ideal of G; that is to say, the kernel of the homomorphism of ZG onto Z which sends each group element to 1. The powers gλ of 𝔤 are defined inductively for ordinals λ by 𝔤λ = 𝔤μ𝔤, if λ = μ + 1, and otherwise. The first ordinal λ for which gλ = 𝔤λ+1 is called the augmentation terminal or simply the terminal of G. For example, if G is either a cyclic group of prime order or else isomorphic with the additive group of rational numbers then gn > 𝔤ω = 0 for all finite n, so that these groups have terminal ω.The groups with finite terminal are well-known and easily described. If G is one such, then every homomorphic image of G must also have finite terminal.

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