scholarly journals Rational points on certain families of symmetric equations

2015 ◽  
Vol 11 (06) ◽  
pp. 1821-1838
Author(s):  
Wade Hindes
Keyword(s):  
Rational Points ◽  
Hasse Principle ◽  
New Family ◽  
Quadratic Twists ◽  
Parity Conjecture ◽  
Hyperelliptic Jacobians

We generalize the work of Dem'janenko and Silverman for the Fermat quartics, effectively determining the rational points on the curves x2m + axm + aym + y2m = b whenever the ranks of some companion hyperelliptic Jacobians are at most one. As an application, we explicitly describe Xd(ℚ) for certain d ≥ 3, where Xd : Td(x) + Td(y) = 1 and Td is the monic Chebychev polynomial of degree d. Moreover, we show how this later problem relates to orbit intersection problems in dynamics. Finally, we construct a new family of genus 3 curves which break the Hasse principle, assuming the parity conjecture, by specifying our results to quadratic twists of x4 - 4x2 - 4y2 + y4 = -6.

scholarly journals The Hardy–Littlewood conjecture and rational points

2014 ◽  
Vol 150 (12) ◽  
pp. 2095-2111 ◽  
Author(s):  
Yonatan Harpaz ◽  
Alexei N. Skorobogatov ◽  
Olivier Wittenberg
Keyword(s):  
Rational Points ◽  
Hasse Principle ◽  
Weak Approximation ◽  
Ground Field ◽  
Formula One ◽  
Littlewood Conjecture ◽  
Pencils Of Conics

AbstractSchinzel’s Hypothesis (H) was used by Colliot-Thélène and Sansuc, and later by Serre, Swinnerton-Dyer and others, to prove that the Brauer–Manin obstruction controls the Hasse principle and weak approximation on pencils of conics and similar varieties. We show that when the ground field is $\mathbb{Q}$ and the degenerate geometric fibres of the pencil are all defined over $\mathbb{Q}$, one can use this method to obtain unconditional results by replacing Hypothesis (H) with the finite complexity case of the generalised Hardy–Littlewood conjecture recently established by Green, Tao and Ziegler.

scholarly journals RATIONAL POINTS ON INTERSECTIONS OF CUBIC AND QUADRIC HYPERSURFACES

2014 ◽  
Vol 14 (4) ◽  
pp. 703-749 ◽  
Author(s):  
T. D. Browning ◽  
R. Dietmann ◽  
D. R. Heath-Brown
Keyword(s):  
Rational Numbers ◽  
Rational Points ◽  
Complete Intersections ◽  
Hasse Principle ◽  
Quadric Hypersurface

We investigate the Hasse principle for complete intersections cut out by a quadric hypersurface and a cubic hypersurface defined over the rational numbers.

FROM SEPARABLE POLYNOMIALS TO NONEXISTENCE OF RATIONAL POINTS ON CERTAIN HYPERELLIPTIC CURVES

2014 ◽  
Vol 96 (3) ◽  
pp. 354-385 ◽  
Author(s):  
NGUYEN NGOC DONG QUAN
Keyword(s):  
Hyperelliptic Curves ◽  
Rational Points ◽  
Hasse Principle ◽  
Sufficient Condition ◽  
Separability Criterion ◽  
Image Position

AbstractWe give a separability criterion for the polynomials of the form $$\begin{equation*} ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e). \end{equation*}$$ Using this separability criterion, we prove a sufficient condition using the Brauer–Manin obstruction under which curves of the form $$\begin{equation*} z^2 = ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e) \end{equation*}$$ have no rational points. As an illustration, using the sufficient condition, we study the arithmetic of hyperelliptic curves of the above form and show that there are infinitely many curves of the above form that are counterexamples to the Hasse principle explained by the Brauer–Manin obstruction.

scholarly journals Rational points and derived equivalence

2021 ◽  
Vol 157 (5) ◽  
pp. 1036-1050
Author(s):  
Nicolas Addington ◽  
Benjamin Antieau ◽  
Katrina Honigs ◽  
Sarah Frei
Keyword(s):  
Rational Point ◽  
Source Code ◽  
Rational Points ◽  
The Other ◽  
Online Version ◽  
Hasse Principle ◽  
Derived Equivalence ◽  
Derived Equivalences ◽  
Closed Fields ◽  
Supplementary Material

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over $\mathbb {Q}$ and $\mathbb {F}_q(t)$ , and conclude with a pair of hyperkähler 4-folds over $\mathbb {Q}$ . The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article.

scholarly journals Rational points on elliptic K3 surfaces of quadratic twist type

2020 ◽  
Author(s):  
Zhizhong Huang
Keyword(s):  
Elliptic Curves ◽  
K3 Surfaces ◽  
Rational Points ◽  
Zariski Topology ◽  
Quadratic Twists ◽  
Quadratic Twist ◽  
Quartic Polynomials

Abstract In studying rational points on elliptic K3 surfaces of the form $$\begin{equation*} f(t)y^2=g(x), \end{equation*}$$ where f, g are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell–Weil rank, and we relate it to the Hilbert property. Applying to surfaces of Cassels–Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.

RATIONAL POINTS ON SHIMURA CURVES AND THE MANIN OBSTRUCTION

2017 ◽  
Vol 230 ◽  
pp. 144-159
Author(s):  
KEISUKE ARAI
Keyword(s):  
Number Fields ◽  
Rational Points ◽  
Previous Article ◽  
Hasse Principle ◽  
Shimura Curves

In a previous article, we proved that Shimura curves have no points rational over number fields under a certain assumption. In this article, we give another criterion of the nonexistence of rational points on Shimura curves and obtain new counterexamples to the Hasse principle for Shimura curves. We also prove that such counterexamples obtained from the above results are accounted for by the Manin obstruction.

scholarly journals Rank Growth of Elliptic Curves in Non-Abelian Extensions

2019 ◽  
Author(s):  
Robert J Lemke Oliver ◽  
Frank Thorne
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Positive Constant ◽  
Rank Zero ◽  
Abelian Extensions ◽  
Quadratic Twists ◽  
Rank One ◽  
Higher Rank ◽  
Parity Conjecture

Abstract Given an elliptic curve $E/\mathbb{Q}$, it is a conjecture of Goldfeld that asymptotically half of its quadratic twists will have rank zero and half will have rank one. Nevertheless, higher rank twists do occur: subject to the parity conjecture, Gouvêa and Mazur constructed $X^{1/2-\epsilon }$ twists by discriminants up to $X$ with rank at least two. For any $d\geq 3$, we build on their work to consider the rank growth of $E$ in degree $d$$S_d$-extensions of $\mathbb{Q}$ with discriminant up to $X$. We prove that there are at least $X^{c_d-\epsilon }$ such fields where the rank grows, where $c_d$ is a positive constant that tends to $1/4$ as $d\to \infty $. Moreover, subject to a suitable parity conjecture, we obtain the same result for fields for which the rank grows by at least two.

scholarly journals Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jie Shu ◽  
Shuai Zhai
Keyword(s):  
Elliptic Curves ◽  
Large Class ◽  
Infinite Order ◽  
Rational Points ◽  
Heegner Points ◽  
Original Curve ◽  
Quadratic Twists

Abstract In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over ℚ {{\mathbb{Q}}} . We prove the existence of explicit infinite families of quadratic twists with analytic ranks 0 and 1 for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank 1. In addition, we show that these families of quadratic twists satisfy the 2-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture.

scholarly journals AN "ANTI-HASSE PRINCIPLE" FOR PRIME TWISTS

2008 ◽  
Vol 04 (04) ◽  
pp. 627-637 ◽  
Author(s):  
PETE L. CLARK
Keyword(s):  
Algebraic Curve ◽  
Hasse Principle ◽  
Shimura Curves ◽  
Positive Density ◽  
Modular Curves ◽  
Quadratic Twists

Given an algebraic curve C/ℚ having points everywhere locally and endowed with a suitable involution, we show that there exists a positive density family of prime quadratic twists of C violating the Hasse principle. The result applies in particular to wN-Atkin–Lehner twists of most modular curves X0(N) and to wp-Atkin–Lehner twists of certain Shimura curves XD+.

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