scholarly journals Uniform Bound for the Number of Rational Points on a Pencil of Curves

2019 ◽  
Author(s):  
Vesselin Dimitrov ◽  
Ziyang Gao ◽  
Philipp Habegger
Keyword(s):  
Number Field ◽  
Rational Points ◽  
Parameter Family ◽  
Uniform Bound ◽  
Torsion Points ◽  
Uniform Bounds ◽  
The Family ◽  
Projective Curves ◽  
Mordell Conjecture

Abstract Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell–Weil rank of the fiber’s Jacobian. Our proof uses Vojta’s approach to the Mordell Conjecture furnished with a height inequality due to the 2nd- and 3rd-named authors. In addition we obtain uniform bounds for the number of torsion points in the Jacobian that lie in each fiber of the family.

Canonical Heights and Preperiodic Points for Certain Weighted Homogeneous Families of Polynomials

2018 ◽  
Vol 2019 (15) ◽  
pp. 4859-4879
Author(s):  
Patrick Ingram
Keyword(s):  
Lower Bound ◽  
Number Field ◽  
Uniform Bounds ◽  
Abc Conjecture ◽  
Canonical Height ◽  
Homogeneous Form ◽  
The Family

Abstract A family f of polynomials over a number field K will be called weighted homogeneous if and only if ft(z) = F(ze, t) for some binary homogeneous form F(X, Y) and some integer e ≥ 2. For example, the family zd + t is weighted homogeneous. We prove a lower bound on the canonical height, of the form \begin{align*} \hat{h}_{f_{t}}(z)\geq \varepsilon \max\!\left\{h_{\mathsf{M}_{d}}(f_{t}), \log|\operatorname{Norm}\mathfrak{R}_{f_{t}}|\right\},\end{align*} for values z ∈ K which are not preperiodic for ft. Here ε depends only on the number field K, the family f, and the number of places at which ft has bad reduction. For suitably generic morphisms $\varphi :\mathbb {P}^{1}\to \mathbb {P}^{1}$, we also prove an absolute bound of this form for t in the image of φ over K (assuming the abc Conjecture), as well as uniform bounds on the number of preperiodic points (unconditionally).

scholarly journals Le complémentaire des puissances -ièmes dans un corps de nombres est un ensemble diophantien

2015 ◽  
Vol 151 (10) ◽  
pp. 1965-1980 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Jan Van Geel
Keyword(s):  
Prime Number ◽  
Rational Points ◽  
Parameter Family ◽  
Symmetric Power ◽  
Symmetric Powers

For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

scholarly journals Uniform bounds on the number of rational points of a family of curves of genus 2

2004 ◽  
Vol 108 (2) ◽  
pp. 241-267 ◽  
Author(s):  
L. Kulesz ◽  
G. Matera ◽  
E. Schost
Keyword(s):  
Rational Points ◽  
Uniform Bounds ◽  
Family Of Curves ◽  
Genus 2

scholarly journals Galois criterion for torsion points of Drinfeld modules

2021 ◽  
pp. 1-12
Author(s):  
Chien-Hua Chen
Keyword(s):  
Maximal Ideal ◽  
Abelian Varieties ◽  
Number Fields ◽  
Rational Points ◽  
Drinfeld Module ◽  
Torsion Point ◽  
Drinfeld Modules ◽  
Torsion Points ◽  
Ideal Formula ◽  
Almost All

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2, but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.

scholarly journals On the Dynamics of the Unified Chaotic System Between Lorenz and Chen Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550122 ◽  
Author(s):  
Jaume Llibre ◽  
Ana Rodrigues
Keyword(s):  
Chaotic System ◽  
Parameter Family ◽  
Algebraic Surfaces ◽  
Hopf Bifurcations ◽  
Global Dynamics ◽  
Differential Systems ◽  
Special Values ◽  
Unified Chaotic System ◽  
Invariant Algebraic Surfaces ◽  
The Family

A one-parameter family of differential systems that bridges the gap between the Lorenz and the Chen systems was proposed by Lu, Chen, Cheng and Celikovsy. The goal of this paper is to analyze what we can say using analytic tools about the dynamics of this one-parameter family of differential systems. We shall describe its global dynamics at infinity, and for two special values of the parameter a we can also describe the global dynamics in the whole ℝ3using the invariant algebraic surfaces of the family. Additionally we characterize the Hopf bifurcations of this family.

scholarly journals p-Torsion points on elliptic curves defined over quadratic fields

1984 ◽  
Vol 96 ◽  
pp. 139-165 ◽  
Author(s):  
Fumiyuki Momose
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Quadratic Fields ◽  
Finite Degree ◽  
Torsion Points

Let p be a prime number and k an algebraic number field of finite degree d. Manin [14] showed that there exists an integer n = n(k,p) (≧0) which satisfies the condition

Double points in families of map germs from ℝ2 to ℝ3

2020 ◽  
pp. 1-18
Author(s):  
J. A. Moya-Pérez ◽  
J. J. Nuño-Ballesteros
Keyword(s):  
Double Point ◽  
Parameter Family ◽  
Milnor Number ◽  
Double Points ◽  
The Family ◽  
Number Formula ◽  
Real Analytic

We show that a 1-parameter family of real analytic map germs [Formula: see text] with isolated instability is topologically trivial if it is excellent and the family of double point curves [Formula: see text] in [Formula: see text] is topologically trivial. In particular, we deduce that [Formula: see text] is topologically trivial when the Milnor number [Formula: see text] is constant.

scholarly journals SYMMETRIC ITINERARY SETS

2019 ◽  
Vol 100 (1) ◽  
pp. 97-108
Author(s):  
MICHAEL F. BARNSLEY ◽  
NICOLAE MIHALACHE
Keyword(s):  
Dynamical Systems ◽  
Parameter Family ◽  
The Family

We consider a one-parameter family of dynamical systems $W:[0,1]\rightarrow [0,1]$ constructed from a pair of monotone increasing diffeomorphisms $W_{i}$ such that $W_{i}^{-1}:$$[0,1]\rightarrow [0,1]$$(i=0,1)$. We characterise the set of symbolic itineraries of $W$ using an attractor $\overline{\unicode[STIX]{x1D6FA}}$ of an iterated closed relation, in the terminology of McGehee, and prove that there is a member of the family for which $\overline{\unicode[STIX]{x1D6FA}}$ is symmetrical.

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