V.29 Rational Points on Curves and the Mordell Conjecture

2010 ◽  
pp. 720-722
Keyword(s):  
Rational Points ◽  
Mordell Conjecture

scholarly journals An Explicit Manin-Dem’janenko Theorem in Elliptic Curves

2018 ◽  
Vol 70 (5) ◽  
pp. 1173-1200 ◽  
Author(s):  
Evelina Viada
Keyword(s):  
Elliptic Curves ◽  
Rational Points ◽  
Sharp Bound ◽  
Explicit Bound ◽  
Minimal Dimension ◽  
Mordell Conjecture

AbstractLet be a curve of genus at least 2 embedded in E1 × … × EN, where the Ei are elliptic curves for i = 1, . . . , N. In this article we give an explicit sharp bound for the Néron–Tate height of the points of contained in the union of all algebraic subgroups of dimension < max(), where is the minimal dimension of a translate (resp. of a torsion variety) containing .As a corollary, we give an explicit bound for the height of the rational points of special curves, proving new cases of the explicit Mordell Conjecture and in particular making explicit (and slightly more general in the CM case) the Manin–Dem’janenko method for curves in products of elliptic curves.

scholarly journals An effective Chabauty–Kim theorem

2019 ◽  
Vol 155 (6) ◽  
pp. 1057-1075 ◽  
Author(s):  
Jennifer S. Balakrishnan ◽  
Netan Dogra
Keyword(s):  
Rational Points ◽  
Higher Rank ◽  
Mordell Conjecture

The Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a non-abelian generalisation of Coleman’s effective Chabauty theorem.

scholarly journals THE EXPLICIT MORDELL CONJECTURE FOR FAMILIES OF CURVES

2019 ◽  
Vol 7 ◽  
Author(s):  
SARA CHECCOLI ◽  
FRANCESCO VENEZIANO ◽  
EVELINA VIADA
Keyword(s):  
Classical Method ◽  
Rational Points ◽  
Computer Search ◽  
General Shape ◽  
Sharp Estimates ◽  
Families Of Curves ◽  
Large Families ◽  
Mordell Conjecture

In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method bases on some explicit and sharp estimates for the height of such rational points, and the bounds are small enough to successfully implement a computer search. As an evidence of the simplicity of its application, we present a variety of explicit examples and explain how to produce many others. In the appendix our method is compared in detail to the classical method of Manin–Demjanenko and the analysis of our explicit examples is carried to conclusion.

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.

scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

scholarly journals Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

2021 ◽  
Author(s):  
Tim Browning ◽  
Shuntaro Yamagishi
Keyword(s):  
Circle Method ◽  
Rational Points ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Waring’S Problem ◽  
Waring Problem ◽  
Waring's Problem ◽  
Main Conjecture ◽  
Higher Dimensional ◽  
Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

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 Distribution of rational points on the real line

1973 ◽  
Vol 15 (2) ◽  
pp. 243-256 ◽  
Author(s):  
T. K. Sheng
Keyword(s):  
Algebraic Number ◽  
Rational Number ◽  
Real Line ◽  
Rational Points ◽  
The Other ◽  
Liouville Numbers ◽  
The Real ◽  
Other Hand ◽  
Image Position

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

scholarly journals Rational points on linear slices of diagonal hypersurfaces

2015 ◽  
Vol 218 ◽  
pp. 51-100
Author(s):  
Jörg Brüdern ◽  
Olivier Robert
Keyword(s):  
Asymptotic Formula ◽  
Rational Points ◽  
Independent Interest ◽  
The Way ◽  
Hardy Littlewood Method

AbstractAn asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest.

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