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Register a new userCubic 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). In a recent paper, for fields K with at least 13 elements, Siksek showed that if S contains a skew pair of K-lines then S(K) can be generated from one point. In this paper we prove the corresponding version of this result for fields K having at least 4 elements, and slightly milder results for #K=2 or 3.
If the $ell$-adic cohomology of a projective smooth variety, defined over a local field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then every model over the ring of integers of $K$ has a $k$-rational point. For $K$ a $p$-adic field, this is math/0405318, Theorem 1.1. If the model $sX$ is regular, one has a congruence $|sX(k)|equiv 1 $ modulo $|k|$ for the number of $k$-rational points 0704.1273, Theorem 1.1. The congruence is violated if one drops the regularity assumption.
In the paper [On superspecial abelian surfaces over finite fields II. J. Math. Soc. Japan, 72(1):303--331, 2020], Tse-Chung Yang and the first two current authors computed explicitly the number $lvert mathrm{SSp}_2(mathbb{F}_q)rvert$ of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field $mathbb{F}_q$ of even degree over the prime field $mathbb{F}_p$. There it was assumed that certain commutative $mathbb{Z}_p$-orders satisfy an etale condition that excludes the primes $p=2, 3, 5$. We treat these remaining primes in the present paper, where the computations are more involved because of the ramifications. This completes the calculation of $lvert mathrm{SSp}_2(mathbb{F}_q)rvert$ in the even degree case. The odd degree case was previous treated by Tse-Chung Yang and the first two current authors in [On superspecial abelian surfaces over finite fields. Doc. Math., 21:1607--1643, 2016]. Along the proof of our main theorem, we give the classification of lattices over local quaternion Bass orders, which is a new input to our previous works.
Let ${mathbb F}_q$ be the finite field with $q=p^k$ elements with $p$ being a prime and $k$ be a positive integer. For any $y, zinmathbb{F}_q$, let $N_s(z)$ and $T_s(y)$ denote the numbers of zeros of $x_1^{3}+cdots+x_s^3=z$ and $x_1^3+cdots+x_{s-1}^3+yx_s^3=0$, 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.
We study the dynamics of maps arising from the composition of two non-commuting involution on a K3 surface. These maps are a particular example of reversible maps, i.e., maps with a time reversing symmetry. The combinatorics of the cycle distribution of two non-commuting involutions on a finite phase space was studied by Roberts and Vivaldi. We show that the dynamical systems of these K3 surfaces satisfy the hypotheses of their results, providing a description of the cycle distribution of the rational points over finite fields. Furthermore, we extend the involutions to include the case where there are degenerate fibers and prove a description of the cycle distribution in this more general situation.
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