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تسجيل مستخدم جديدCubic surfaces with one rational line over finite fields
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Jenny Cooley
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Let Fq be a finite field with q=8 or q at least 16. Let S be a smooth cubic surface defined over Fq containing at least one rational line. We use a pigeonhole principle to prove that all the rational points on S are generated via tangent and secant operations from a single point.
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Let S be a smooth cubic surface defined over a field K. As observed by Segre and Manin, there is a secant and tangent process on S that generates new K-rational points from old. It is natural to ask for the size of a minimal generating set for S(K).
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3+yx_s^3=0$, respectively. Gauss proved that if $q=p, pequiv1pmod3$ and $y$ is non-cubic, then $T_3(y)=p^2+frac{1}{2}(p-1)(-c+9d)$, where $c$ and $d$ are uniquely determined by $4p=c^2+27d^2,~cequiv 1 pmod 3$ except for the sign of $d$. In 1978, Chowla, Cowles and Cowles determined the sign of $d$ for the case of $2$ being a non-cubic element of ${mathbb F}_p$. But the sign problem is kept open for the remaining case of $2$ being cubic in ${mathbb F}_p$. In this paper, we solve this sign problem by determining the sign of $d$ when $2$ is cubic in ${mathbb F}_p$. Furthermore, we show that the generating functions $sum_{s=1}^{infty} N_{s}(z) x^{s}$ and $sum_{s=1}^{infty} T_{s}(y)x^{s}$ are rational functions for any $z, yinmathbb F_q^*:=mathbb F_qsetminus {0}$ with $y$ being non-cubic over ${mathbb F}_q$ and also give their explicit expressions. This extends the theorem of Myerson and that of Chowla, Cowles and Cowles.
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