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.

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 Arithmetic E8 Lattices with Maximal Galois Action

2009 ◽  
Vol 12 ◽  
pp. 144-165 ◽  
Author(s):  
Anthony Várilly-Alvarado ◽  
David Zywina
Keyword(s):  
Elliptic Curves ◽  
Weyl Group ◽  
Number Fields ◽  
Picard Group ◽  
Del Pezzo Surfaces ◽  
Full Group ◽  
Group Of Automorphisms ◽  
Intersection Pairing ◽  
Galois Action ◽  
Del Pezzo

AbstractWe construct explicit examples of E8 lattices occurring in arithmetic for which the natural Galois action is equal to the full group of automorphisms of the lattice, i.e., the Weyl group of E8. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E8 and have maximal Galois action.Our main objects of study are del Pezzo surfaces of degree 1 over number fields. The geometric Picard group, considered as a lattice via the negative of the intersection pairing, contains a sublattice isomorphic to E8. We construct examples of such surfaces for which the action of Galois on the geometric Picard group is maximal.

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 On the error term of a lattice counting problem, II

2020 ◽  
Vol 16 (05) ◽  
pp. 1153-1160
Author(s):  
Olivier Bordellès
Keyword(s):  
Asymptotic Formula ◽  
Riemann Hypothesis ◽  
Error Term ◽  
Exponential Sums ◽  
Main Term ◽  
Counting Problem ◽  
Lattice Problem

Under the Riemann Hypothesis, we improve the error term in the asymptotic formula related to the counting lattice problem studied in a first part of this work. The improvement comes from the use of Weyl’s bound for exponential sums of polynomials and a device due to Popov allowing us to get an improved main term in the sums of certain fractional parts of polynomials.

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