scholarly journals A positive proportion of Thue equations fail the integral Hasse principle

2019 ◽  
Vol 141 (2) ◽  
pp. 283-307 ◽  
Author(s):  
Shabnam Akhtari ◽  
Manjul Bhargava
Keyword(s):  
Hasse Principle ◽  
Positive Proportion

Thue equations that simultaneously fail the Hasse principle

2021 ◽  
pp. 1-10
Author(s):  
PALOMA BENGOECHEA
Keyword(s):  
Positive Integer ◽  
Hasse Principle ◽  
Fixed Degree ◽  
Binary Forms ◽  
Absolute Value ◽  
Positive Proportion ◽  
The Absolute

Abstract We refine a previous construction by Akhtari and Bhargava so that, for every positive integer m, we obtain a positive proportion of Thue equations F(x, y) = h that fail the integral Hasse principle simultaneously for every positive integer h less than m. The binary forms F have fixed degree ≥ 3 and are ordered by the absolute value of the maximum of the coefficients.

scholarly journals Brauer–Manin obstruction and families of generalised Châtelet surfaces over number fields

2019 ◽  
Vol 15 (02) ◽  
pp. 289-308
Author(s):  
Francesca Balestrieri
Keyword(s):  
Sufficient Conditions ◽  
Number Fields ◽  
Specific Form ◽  
Hasse Principle ◽  
Galois Extensions ◽  
Positive Proportion ◽  
Odd Degree

Over infinitely many number fields [Formula: see text] (including all finite Galois extensions [Formula: see text] of odd degree unramified at 2), we give general sufficient conditions in order for the generalised Châtelet surfaces [Formula: see text] over [Formula: see text] associated to the normic equation [Formula: see text], where [Formula: see text] has a specific form and [Formula: see text] is even and arbitrarily large, to have the property that [Formula: see text] but [Formula: see text]. We also give general sufficient conditions in order for the generalised Châtelet surfaces [Formula: see text] over [Formula: see text] of the same form as above to have the property that [Formula: see text] and [Formula: see text]. As an application, we prove that, for a certain family of generalised Châtelet surfaces over [Formula: see text], a positive proportion (but not 100[Formula: see text]) of its members exhibits a violation of the Hasse principle explained by the Brauer–Manin obstruction.

scholarly journals A positive proportion of Hasse principle failures in a family of Châtelet surfaces

2019 ◽  
Vol 15 (06) ◽  
pp. 1237-1249
Author(s):  
Nick Rome
Keyword(s):  
Asymptotic Formula ◽  
Hasse Principle ◽  
Positive Proportion

We investigate a family of Châtelet surfaces over [Formula: see text] and develop an asymptotic formula for the frequency of Hasse principle failures. We show that a positive proportion (roughly 23.7%) of such surfaces fails the Hasse principle, by building on previous work of De La Bretèche and Browning.

On the Hasse principle for cubic surfaces

Mathematika ◽  
1966 ◽  
Vol 13 (2) ◽  
pp. 111-120 ◽  
Author(s):  
J. W. S. Cassels ◽  
M. J. T. Guy
Keyword(s):  
Hasse Principle ◽  
Cubic Surfaces

scholarly journals ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS

2008 ◽  
Vol 78 (3) ◽  
pp. 431-436 ◽  
Author(s):  
XUE-GONG SUN ◽  
JIN-HUI FANG
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Positive Proportion ◽  
Covering System

AbstractErdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

Sign in / Sign up

Export Citation Format

Share Document