Details
| Original language | English |
|---|---|
| Pages (from-to) | 755-772 |
| Number of pages | 18 |
| Journal | Quarterly Journal of Mathematics |
| Volume | 72 |
| Issue number | 3 |
| Early online date | 28 Nov 2020 |
| Publication status | Published - Sept 2021 |
Abstract
In studying rational points on elliptic K3 surfaces of the form $$\begin{equation∗} f(t)y^2=g(x), \end{equation∗}$$ where f, g are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell-Weil rank. We prove a necessary and sufficient condition for the Zariski density of rational points by using this condition, and we relate it to the Hilbert property. Applying to surfaces of Cassels-Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Quarterly Journal of Mathematics, Vol. 72, No. 3, 09.2021, p. 755-772.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Rational Points on Elliptic K3 Surfaces of Quadratic Twist Type
AU - Huang, Zhizhong
PY - 2021/9
Y1 - 2021/9
N2 - In studying rational points on elliptic K3 surfaces of the form $$\begin{equation∗} f(t)y^2=g(x), \end{equation∗}$$ where f, g are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell-Weil rank. We prove a necessary and sufficient condition for the Zariski density of rational points by using this condition, and we relate it to the Hilbert property. Applying to surfaces of Cassels-Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.
AB - In studying rational points on elliptic K3 surfaces of the form $$\begin{equation∗} f(t)y^2=g(x), \end{equation∗}$$ where f, g are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell-Weil rank. We prove a necessary and sufficient condition for the Zariski density of rational points by using this condition, and we relate it to the Hilbert property. Applying to surfaces of Cassels-Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.
UR - http://www.scopus.com/inward/record.url?scp=85116468192&partnerID=8YFLogxK
U2 - 10.1093/qmath/haaa044
DO - 10.1093/qmath/haaa044
M3 - Article
AN - SCOPUS:85116468192
VL - 72
SP - 755
EP - 772
JO - Quarterly Journal of Mathematics
JF - Quarterly Journal of Mathematics
SN - 0033-5606
IS - 3
ER -