scholarly journals On the moments of torsion points modulo primes and their applications

2020 ◽  
pp. 1-17
Author(s):  
Amir Akbary ◽  
Peng-Jie Wong
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Residue Field ◽  
Density Theorem ◽  
Polynomial Equations ◽  
Absolute Galois Group ◽  
Torsion Points ◽  
Chebotarev Density Theorem ◽  
System Of Polynomial Equations ◽  
Dimension One

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.

scholarly journals ON THE DISTRIBUTION OF TORSION POINTS MODULO PRIMES

2012 ◽  
Vol 86 (2) ◽  
pp. 339-347 ◽  
Author(s):  
YEN-MEI J. CHEN ◽  
YEN-LIANG KUAN
Keyword(s):  
Positive Integer ◽  
Algebraic Group ◽  
Number Field ◽  
Divisor Function ◽  
Good Reduction ◽  
Torsion Points ◽  
Average Value ◽  
Finite Set ◽  
Dimension One

AbstractLet $\Bbb A$ be a commutative algebraic group defined over a number field K. For a prime ℘ in K where $\Bbb A$ has good reduction, let N℘,n be the number of n-torsion points of the reduction of $\Bbb A$ modulo ℘ where n is a positive integer. When $\Bbb A$ is of dimension one and n is relatively prime to a fixed finite set of primes depending on $\Bbb A_{/K}$, we determine the average values of N℘,n as the prime ℘ varies. This average value as a function of n always agrees with a divisor function.

A note on uniform definability and minimal fields of definition

2000 ◽  
Vol 65 (2) ◽  
pp. 817-821
Author(s):  
Rahim Moosa
Keyword(s):  
Residue Field ◽  
Field Extension ◽  
Closed Field ◽  
Polynomial Equations ◽  
Hilbert Schemes ◽  
Prime Field ◽  
Field Of Definition ◽  
Flat Family ◽  
System Of Polynomial Equations ◽  
Image Position

Let k ⊂ K be a field extension, where K is an algebraically closed field of any characteristic and k is the prime field. Recall the following property of Hilbert Schemes (see, for example, [1], Proposition 1.16): Suppose ⊂ × S is a flat family of closed subschemes of parametrised by a scheme S/k. Then for every closed subscheme Z ⊂ in , if [Z] denotes the Hilbert point of Z in Hilb() then the residue field of Hilb() at [Z] is the minimal field of definition for Z. Intuitively, this says that as a family parametrised by Hilb(), each fibre of lies above a point whose “co-ordinates” generate its minimal field of definition.In the following note we are concerned with a more naive form of the above situation, for which we wish to give an elementary account. Suppose ϕ(x,y) is a system of polynomial equations over k (in variables x = (x1,…, xm) and parameters y = (y1, …, yn)), such thatis a family of (possibly reducible) affine varieties in Km. Each nonempty member of this family has a unique minimal field of definition. The question arises as to whether it is possible to express this family of varieties using parameters that come, pointwise, from these minimal fields of definition. That is, is there a system of polynomial equations ψ(x, z) over k, such that each ψ(x, b) with b ∈ KN is of the form Va for some a ∈ Kn; and such that each Va ∈ is defined by ψ(x, b) for some b ∈ KN whose coordinates generate the minimal field of definition for Va? Moreover, we would like b to be obtained definably from a.

scholarly journals A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations

2009 ◽  
Vol 47 (5) ◽  
pp. 3608-3623 ◽  
Author(s):  
Daniel J. Bates ◽  
Jonathan D. Hauenstein ◽  
Chris Peterson ◽  
Andrew J. Sommese
Keyword(s):  
Solution Set ◽  
Polynomial Equations ◽  
Local Dimension ◽  
System Of Polynomial Equations

scholarly journals Bivariate systems of polynomial equations with roots of high multiplicity

2021 ◽  
Author(s):  
I. Nikitin
Keyword(s):  
High Multiplicity ◽  
Polynomial Equations ◽  
Support Sets ◽  
Multiplicity Formula ◽  
System Of Polynomial Equations ◽  
Systems Of Polynomial Equations

Given a bivariate system of polynomial equations with fixed support sets [Formula: see text] it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity [Formula: see text] for all [Formula: see text] in the range [Formula: see text], where [Formula: see text] is the set of all integral vectors that shift B to a subset of [Formula: see text]. As an application, we classify all pairs [Formula: see text] such that the system supported at [Formula: see text] does not have a solution of multiplicity higher than [Formula: see text].

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 An explicit Chebotarev density theorem under GRH

2019 ◽  
Vol 200 ◽  
pp. 441-485 ◽  
Author(s):  
Loïc Grenié ◽  
Giuseppe Molteni
Keyword(s):  
Density Theorem ◽  
Chebotarev Density Theorem

scholarly journals Shintani descent for algebraic groups and almost characters of unipotent groups

2016 ◽  
Vol 152 (8) ◽  
pp. 1697-1724 ◽  
Author(s):  
Tanmay Deshpande
Keyword(s):  
Finite Field ◽  
Algebraic Group ◽  
Algebraic Groups ◽  
Unipotent Group ◽  
Connected Component ◽  
Unipotent Groups ◽  
Character Sheaves ◽  
Treat All ◽  
Trace Of Frobenius

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_{q}$. For this, it is essential to treat all the pure inner $\mathbb{F}_{q}$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the ‘trace of Frobenius’ functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko and Drinfeld’s theory of character sheaves on neutrally unipotent groups are in fact positive integral, confirming a conjecture due to Drinfeld.

scholarly journals Monothetic algebraic groups

2007 ◽  
Vol 82 (3) ◽  
pp. 315-324 ◽  
Author(s):  
Giovanni Falcone ◽  
Peter Plaumann ◽  
Karl Strambach
Keyword(s):  
Algebraic Group ◽  
Cyclic Subgroup ◽  
Algebraic Groups ◽  
Arbitrary Field

AbstractWe call an algebraic group monothetic if it possesses a dense cyclic subgroup. For an arbitrary field k we describe the structure of all, not necessarily affine, monothetic k-groups G and determine in which cases G has a k-rational generator.

Analysis of secret sharing schemes based on Nielsen transformations

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Matvei Kotov ◽  
Dmitry Panteleev ◽  
Alexander Ushakov
Keyword(s):  
Computer Algebra ◽  
Secret Sharing ◽  
Computer Algebra System ◽  
Cryptographic Protocols ◽  
Secret Sharing Schemes ◽  
Polynomial Equations ◽  
Polynomial Size ◽  
Security Properties ◽  
System Of Polynomial Equations

Abstract We investigate security properties of two secret-sharing protocols proposed by Fine, Moldenhauer, and Rosenberger in Sections 4 and 5 of [B. Fine, A. Moldenhauer and G. Rosenberger, Cryptographic protocols based on Nielsen transformations, J. Comput. Comm. 4 2016, 63–107] (Protocols I and II resp.). For both protocols, we consider a one missing share challenge. We show that Protocol I can be reduced to a system of polynomial equations and (for most randomly generated instances) solved by the computer algebra system Singular. Protocol II is approached using the technique of Stallings’ graphs. We show that knowledge of {m-1} shares reduces the space of possible values of a secret to a set of polynomial size.

scholarly journals Representations of Algebraic Groups

1963 ◽  
Vol 22 ◽  
pp. 33-56 ◽  
Author(s):  
Robert Steinberg
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Vector Spaces ◽  
Irreducible Representations ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Field ◽  
Infinite Degree ◽  
Semisimple Algebraic Groups

Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of corresponding finite simple groups. (Each algebraic group is assumed to be defined over a universal field which is algebraically closed and of infinite degree of transcendence over the prime field, and all of its representations are assumed to take place on vector spaces over this field.)

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