scholarly journals Inhomogeneous cubic congruences and rational points on del Pezzo surfaces

2013 ◽  
Vol 2013 (680) ◽  
Author(s):  
Stephan Baier ◽  
Tim D. Browning
Keyword(s):  
Rational Points ◽  
Del Pezzo Surfaces ◽  
Del Pezzo

scholarly journals Counting rational points on quartic del Pezzo surfaces with a rational conic

2018 ◽  
Vol 373 (3-4) ◽  
pp. 977-1016
Author(s):  
T. D. Browning ◽  
E. Sofos
Keyword(s):  
Rational Points ◽  
Del Pezzo Surfaces ◽  
Del Pezzo ◽  
Counting Rational Points

scholarly journals Equivariant Compactifications of Two-Dimensional Algebraic Groups

2014 ◽  
Vol 58 (1) ◽  
pp. 149-168 ◽  
Author(s):  
Ulrich Derenthal ◽  
Daniel Loughran
Keyword(s):  
Homogeneous Spaces ◽  
Algebraic Groups ◽  
Rational Points ◽  
Del Pezzo Surfaces ◽  
Direct Products ◽  
Two Dimensional ◽  
Del Pezzo ◽  
Manin’S Conjecture ◽  
Transitive Actions

AbstractWe classify generically transitive actions of semi-direct products on ℙ2. Motivated by the program to study the distribution of rational points on del Pezzo surfaces (Manin's conjecture), we determine all (possibly singular) del Pezzo surfaces that are equivariant compactifications of homogeneous spaces for semi-direct products .

scholarly journals RATIONAL POINTS ON CERTAIN DEL PEZZO SURFACES OF DEGREE ONE

2008 ◽  
Vol 50 (3) ◽  
pp. 557-564 ◽  
Author(s):  
MACIEJ ULAS
Keyword(s):  
Rational Points ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Zariski Topology ◽  
Del Pezzo Surface ◽  
Del Pezzo

AbstractLet$f(z)=z^5+az^3+bz^2+cz+d \in \Z[z]$and let us consider a del Pezzo surface of degree one given by the equation$\cal{E}_{f}\,{:}\,x^2-y^3-f(z)=0$. In this paper we prove that if the set of rational points on the curveEa,b:Y2=X3+ 135(2a−15)X−1350(5a+ 2b− 26) is infinite then the set of rational points on the surface ϵfis dense in the Zariski topology.

scholarly journals Density of rational points on del Pezzo surfaces of degree one

2014 ◽  
Vol 261 ◽  
pp. 154-199 ◽  
Author(s):  
Cecília Salgado ◽  
Ronald van Luijk
Keyword(s):  
Rational Points ◽  
Del Pezzo Surfaces ◽  
Del Pezzo

scholarly journals Counting rational points on del Pezzo surfaces with a conic bundle structure

2014 ◽  
Vol 163 (3) ◽  
pp. 271-298 ◽  
Author(s):  
Tim Browning ◽  
Michael Swarbrick Jones
Keyword(s):  
Rational Points ◽  
Del Pezzo Surfaces ◽  
Conic Bundle ◽  
Del Pezzo ◽  
Counting Rational Points ◽  
Bundle Structure

Elliptic curves with square-free Δ

2016 ◽  
Vol 12 (03) ◽  
pp. 737-764 ◽  
Author(s):  
Stephan Baier
Keyword(s):  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Error Term ◽  
Exponential Sums ◽  
Rational Points ◽  
Del Pezzo Surfaces ◽  
Explicit Evaluation ◽  
Del Pezzo

Under the Riemann Hypothesis for Dirichlet [Formula: see text]-functions, we improve on the error term in a smoothed version of an estimate for the density of elliptic curves with square-free [Formula: see text], where [Formula: see text] is the discriminant, by the author and Browning [Inhomogeneous cubic congruences and rational points on Del Pezzo surfaces, J. Reine Angew. Math. 680 (2013) 69–151]. To achieve this improvement, we elaborate on our methods for counting weighted solutions of inhomogeneous cubic congruences to powerful moduli. The novelty lies in going a step further in the explicit evaluation of complete exponential sums and saving a factor by averaging over the moduli.

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