Rational points on certain del Pezzo surfaces of degree one

Details Rational points on certain del Pezzo surfaces of degree one Rational points on certain del Pezzo surfaces of degree one

2009-01-17

arxiv.org/abs/0901.2658v1

Mathematics

Number Theory

8 pages. Published in Glasgow Mathematical Journal

Scientific paper

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 note we prove that if the set of rational points on the curve $E_{a,
b}:Y^2=X^3+135(2a-15)X-1350(5a+2b-26)$ is infinite, then the set of rational
points on the surface $\cal{E}_{f}$ is dense in the Zariski topology.

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