• Corpus ID: 207847641

# Geometry of the del Pezzo surface y^2=x^3+Am^6+Bn^6

@article{Desjardins2019GeometryOT,
title={Geometry of the del Pezzo surface y^2=x^3+Am^6+Bn^6},
journal={arXiv: Number Theory},
year={2019}
}
• Published 7 November 2019
• Mathematics
• arXiv: Number Theory
In this paper, we give an effective and efficient algorithm which on input takes non-zero integers $A$ and $B$ and on output produces the generators of the Mordell-Weil group of the elliptic curve over $\mathbb{Q}(t)$ given by an equation of the form $y^2=x^3+At^6+B$. Our method uses the correspondence between the 240 lines of a del Pezzo surface of degree 1 and the sections of minimal Shioda height on the corresponding elliptic surface over $\overline{\mathbb{Q}}$. For most rational elliptic…
2 Citations

## Figures and Tables from this paper

Root number of twists of an elliptic curve
We give an explicit description of the behaviour of the root number in the family given by the twists of an elliptic curve $E$ by the rational values of a polynomial $f(T)$. In particular, we give a
Density of rational points on a family of del Pezzo surfaces of degree one
• Mathematics
• 2021
Let k be an infinite field of characteristic 0, and X a del Pezzo surface of degree d with at least one k-rational point. Various methods from algebraic geometry and arithmetic statistics have shown

## References

SHOWING 1-10 OF 23 REFERENCES
RATIONAL POINTS ON CERTAIN DEL PEZZO SURFACES OF DEGREE ONE
• M. Ulas
• Mathematics
Glasgow Mathematical Journal
• 2008
Abstract Let $f(z)=z^5+az^3+bz^2+cz+d \in \Z[z]$ 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
Density of rational points on del Pezzo surfaces of degree one
• Mathematics
• 2012
We state conditions under which the set S(k) of k-rational points on a del Pezzo surface S of degree 1 over an infinite field k of characteristic not equal to 2 or 3 is Zariski dense. For example, it
Rational points on certain elliptic surfaces
Let Ef : y 2 = x 3 +f(t)x, where f ∈ Q(t) \ Q, and let us assume that degf ≤ 4. In this paper we prove that if degf ≤ 3, then there exists a rational base change t 7→ '(t) such that there is a
On the density of rational points on rational elliptic surfaces
Let $\mathscr{E}\rightarrow\mathbb{P}^1_\mathbb{Q}$ be a non-trivial rational elliptic surface over $\mathbb{Q}$ with base $\mathbb{P}^1_\mathbb{Q}$ (with a section). We conjecture that any
On the unirationality of del Pezzo surfaces of degree 2
• Mathematics, Computer Science
J. Lond. Math. Soc.
• 2014
It is shown that all except possibly three explicit del Pezzo surfaces of degree two are unirational over k, where k is a finite field.
Weak approximation on del Pezzo surfaces of degree 1
We study del Pezzo surfaces of degree 1 of the form w2=z3+Ax6+By6in the weighted projective space , where k is a perfect field of characteristic not 2 or 3 and A,Bk*. Over a number field, we
Cubic forms; algebra, geometry, arithmetic
CH-Quasigroups and Moufang Loops. Classes of Points on Cubic Hypersurfaces. Two-Dimensional Birational Geometry. The Twenty-Seven Lines. Minimal Cubic Surfaces. The Brauer-Grothendieck Group.
Heights and the specialization map for families of abelian varieties.
Let C be a non-singular projective curve, and let A — > C be a (flat) family of abelian varieties, all defmed over a global field K. There are three natural height functions associated to such a
Root numbers and parity of ranks of elliptic curves
• Mathematics
• 2011
Abstract The purpose of the paper is to complete several global and local results concerning parity of ranks of elliptic curves. Primarily, we show that the Shafarevich–Tate conjecture implies the
Density of rational points on isotrivial rational elliptic surfaces
For a large class of isotrivial rational elliptic surfaces (with section), we show that the set of rational points is dense for the Zariski topology, by carefully studying variations of root numbers