D. R. Heath-Brown
The Cayley cubic surface is given by the equation sum_{i=1}^4 X_i^{-1}=0. We show that the number of non-trivial primitive integer points of size at most B is of exact order B(log B)^6, as predicted by Manin's conjecture.
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We count rational points of bounded height on the Cayley ruled cubic surface and interpret the result in the context of general conjectures due to Batyrev and Tschinkel.
We investigate the density of rational points on Cayley's cubic surface whose coordinates have few prime factors. The key tools used are the circle method and universal torsors.
In this note we study the integer solutions of Cayley's cubic equation. We find infinite families of solutions built from recurrence relations. We use these solutions to solve certain general Pell equations. We also show the similarities and differences to Markov numbers. In particular we introduce new formulae for the solutions to Cayley's cubic equation in analogy with Markov numbers and discuss their distinctions.
A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Such Diophantine sets have been used to construct high rank elliptic curves. Here I demonstrate a link with Cayley's hyperdeterminants which provides a fruitful generalisation of Diophantine quadruples.
Cubic forms over imaginary quadratic number fields and pairs of rational cubic forms
该研究拓展Heath-Brown的有理三次型成果，证明14变量以上虚二次数域三次型非平凡表示零，推导得s≥627时有理三次型对有非平凡解，还证明33变量以上有理三次超曲面含有理直线、对应三次型有殆素数解。
Building on recent work of Bhargava--Elkies--Schnidman and Kriz--Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.
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.
We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a combination of the "determinant method" with an m-descent on the curve.
We classify the possible torsion structures of rational elliptic curves over cubic fields. Along the way we find a previously unknown torsion structure over a cubic field, $\Z /21 \Z$, which corresponds to a sporadic point on $X_1(21)$ of degree 3, which is the lowest possible degree of a sporadic point on a modular curve $X_1(n)$.
Using essentially only algebra, we give a proof that a cubic rational function over $\mathbb{C}$ with real critical points is equivalent to a real rational function. We also show that the natural generalization to $\mathbb{Q}_p$ fails for all $p$.
Rational lines on cubic hypersurfaces II
本文研究了三次超曲面上的有理直线问题，通过引入二次型线性空间和虚二次数域上的三次型结果，将三次超曲面的有理直线存在下界从33降低至29。
For any finite field k of characteristic exceeding 3, the Hasse principle and weak approximation is established for non-singular cubic hypersurfaces X over the function field k(t), provided that X has dimension at least 6.
Let $E$ be an elliptic curve defined over $\Q$, and let $G$ be the torsion group $E(K)_{tors}$ for some cubic field $K$ which does not occur over $\Q$. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) $G$ can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves $E/\Q$ together with cubic fields $K$ so that $G= E(K)_{tors}$.
We show that if $X\subseteq \mathbb{P}^{n-1}$, defined over $\mathbb{Q}$ by a cubic form that splits off two forms, with $n\geq 11$, then $X(\mathbb{Q})$ is non-empty. The same holds for an $(m_1,m_2)$-form with $m_1\geq 4$ and $m_2\geq 5$.
Let n be an odd number and F an imaginary quadratic field with odd discriminant. We show that there exists infinitely many cubic fields K such that the class number of K is divisible by n and the Galois closure of K contains F.
Given a smooth cubic hypersurface $X$ over a finite field of characteristic greater than 3 and two generic points on $X$, we use a function field analogue of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number of degree $d$ rational curves on $X$ passing through those two points. We use this to deduce the dimension and irreducibility of the moduli space parametrising such curves, for large enough $d$.
We show that any smooth projective cubic hypersurface of dimension at least $29$ over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley. We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point.
A perfect cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. The existence of such cuboids is neither proved, nor disproved. A rational perfect cuboid is a natural companion of a perfect cuboid absolutely equivalent to the latter one. Its edges and face diagonals are rational numbers, while its space diagonal is equal to unity. Recently, based on a symmetry reduction, it was shown that edges of a rational perfect cuboid are roots of a certain cubic equation with rational coefficients depending on two rational parameters. F…
We investigate the density of rational points on the Fermat cubic surface and the Cayley cubic surface whose coordinates have few prime factors. The key tools used are the weighted sieve, the circle method and universal torsors.
On $ p-$Rationality of Cubic and Quartic Number Fields
本文提出新的判据，通过三次递推序列的p-可除性判断某些复三次数字段的p-理性性，并探讨广义abc猜想与p-理性性的关系，获得满足Greenberg广义猜想的显式数域。
We show that the proportion of plane cubic curves over ${\mathbb Q}_p$ that have a ${\mathbb Q}_p$-rational point is a rational function in $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. As a consequence, we obtain the density of plane cubic curves over ${\mathbb Q}$ that have points everywhere locally; numerically, this density is shown to be $\approx 97.3\%$.
On the density of rational lines on diagonal cubic hypersurfaces, II
本文通过改进小弧均值估计并利用两个维度上的变量平移技巧，建立了18个或更多变量对角三次超曲面上有理线数量的预期渐近公式。
Rational Points on Diagonal Cubic Surfaces
本文在假设椭圆曲线的Tate-Shafarevich群有限的前提下，证明了特定对角三次曲面存在有理点。
In 1958 E. Lehmer found an explicit description of those primes p for which a given prime q is a cubic residue. In this paper we demonstrate that a similar result may be obtained for cubic nonresidues, yielding a cubic character for fixed p that provides an effective means for ascertaining whether or not an arbitrary integer c is a cubic residue modulo p. As an illustration of this technique, we determine whether 1982 is a cubic residue modulo the 131-digit prime p=(3^19+5^82)/4, a question which is essentially impossible to answer with Lehmer's original criterion.
How often does a cubic hypersurface have a rational point?
本文借助三次超曲面平均满足 Hasse 原理的新结果，按三次型高度计算 $n\geq4$ 时含有理点的超曲面比例，并给出 $n=3$ 三次曲面的精确猜想。
On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. We prove that the number of prime terms in the sequence is uniformly bounded. When the rational point is the image of another rational point under a certain 3-isogeny, all terms beyond the first fail to be primes.
In this paper, the Hermite problem has been approached finding a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In other words, the problem of writing cubic irrationals as a periodic sequence of rational or integer numbers has been solved. In particular, a periodic multidimensional continued fraction (with pre--period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences. This multidimensional continued fraction …
Binary Cubic Forms and Rational Cube Sum Problem
本文利用整二元三次型研究有理数立方和问题，无条件证明了在特定剩余类中存在无穷多个素数可表示为两个有理数的立方和。
Rational solutions of pairs of diagonal equations, one cubic and one quadratic
本文通过获得指数和 $αx^3+βx^2$ 的 32/3 阶矩的最优估计，在局部可解性假设下证明变量数超过 10 时一对三次和二次对角方程存在非平凡整数解。
Prescribed Ideal Classes in Ennola Cubic Fields and Rational Points on Associated Elliptic Curves
本文构造了具有指定理想类阶的Ennola三次域，并通过椭圆曲线和Pell方程获得多个独立有理点及高秩纤维。
On the density of rational lines on diagonal cubic hypersurfaces
本文通过多维变量移位和剪枝论证，结合Parsell-Vinogradov系统的最新进展，证明了当s≥19时，对角三次超曲面∑_{i=1}^s c_i x_i^3=0上有理线的渐近估计，改进了此前s≥21的界限。
Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(3^\infty)$ be the compositum of all cubic extensions of $\mathbb{Q}$. In this article we show that the torsion subgroup of $E(\mathbb{Q}(3^\infty))$ is finite and determine 20 possibilities for its structure, along with a complete description of the $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves, and a complete list of $j$-inv…
We prove a lower bound that agrees with Manin's prediction for the number of rational points of bounded height on the Fermat cubic surface. As an application we provide a simple counterexample to Manin's conjecture over the rationals.
We show that the count of rational points by de la Bretèche, Browning and Salberger on the Cayley ruled cubic surface extends to all non-normal integral hypercubics which are not cones.
We investigate the Hasse principle for complete intersections cut out by a quadric and cubic hypersurface defined over the rational numbers.
Rational approximation on cubic hypersurfaces
针对数域上存在k-有理直线的光滑三次超曲面，研究其有理点的逼近常数计算问题，完成了对应常数的推导并验证了Coba猜想，为相关代数几何与数论研究提供了支撑。
Let X be a geometrically integral projective cubic hypersurface defined over the rationals, with dimension D and singular locus of dimension at most D-4. For any ε>0, we show that X contains O(B^{D+ε}) rational points of height at most B. The implied constant in this estimate depends upon the choice of εand the coefficients of the cubic form defining X.
We report on our project to construct non-singular cubic surfaces over $\bbQ$ with a rational line. Our method is to start with degree 4 Del Pezzo surfaces in diagonal form. For these, we develop an explicit version of Galois descent.
We prove that any smooth cubic surface defined over any number field satisfies the lower bound predicted by Manin's conjecture possibly after an extension of small degree.
We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A_5+A_1.
Let k be a finite field with characteristic exceeding 3. We prove that the space of rational curves of fixed degree on any smooth cubic hypersurface over k with dimension at least 11 is irreducible and of the expected dimension.
We show that the number of non-trivial rational points of height at most $B$, that lie on the cubic surface $x_1x_2x_3=x_4(x_1+x_2+x_3)^2$, has order of magnitude $B(\log B)^6$. This agrees with the Manin conjecture.
These are the substantially expanded notes of the lectures of JK at the summer school "Higher-Dimensional Geometry over Finite Fields" in Göttingen, June 2007. The first part gives an overview of the methods. The main new result is the construction of rational curves passing through a given collection of points on smooth cubic hypersurfaces over finite fields.
Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over the rationals. In this paper it is shown that X contains rational points provided that the cubic form defining X can be written as the sum of two forms that share no common variables. ----- This paper features an appendix "Groupe de Brauer non ramifié des hypersurfaces cubiques singulières (d'après P. Salberger)", by J.-L. Colliot-Thĺène.
On the rational points in conics of a cubic surface
本文通过推广的全局行列式方法，研究了三次曲面中圆锥提供的有理点的有界高度上界问题。
For given non-zero integers a,b,q we investigate the density of integer solutions (x,y) to the binary cubic congruence ax^2+by^3=0 (mod q). We use this to establish the Manin conjecture for a singular del Pezzo surface of degree 2 defined over the rationals and to examine the distribution of elliptic curves with square-free discriminant.
We use a global version of Heath-Brown's $p-$adic determinant method developed by Salberger to give upper bounds for the number of rational points of height at most $B$ on non-singular cubic curves defined over $\mathbb{Q}$. The bounds are uniform in the sense that they only depend on the rank of the corresponding Jacobian.
We establish estimates for the number of solutions of certain affine congruences. These estimates are then used to prove Manin's conjecture for a cubic surface split over Q and whose singularity type is D_4. This improves on a result of Browning and answers a problem posed by Tschinkel.
Rational points on even dimensional Fermat cubics
本文利用低次多项式的显式有理参数化，证明偶数维 Fermat 三次超曲面在特征非三的任意域上有理，并导出数域有理点估计及一类二次—三次 Cremona 对应。
Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a cubic number field. In particular, We study the number of cubic number fields K such that E(Q)_tors\neq E(K)_tors.
This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this particular case, we give a description of the possible torsion growth over cubic fields and a completely explicit description of this growth in terms of some invariants attached to a given elliptic curve.
A conjecture of Manin predicts the distribution of K-rational points on certain algebraic varieties defined over a number field K. In recent years, a method using universal torsors has been successfully applied to several hard special cases of Manin's conjecture over the field Q of rational numbers. Combining this method with techniques developed by Schanuel, we give a proof of Manin's conjecture over arbitrary number fields for the singular cubic surface S given by the equation w^3 = x y z.
We study and partially classify cubic rational expressions $g(x)/h(x)$ over a finite field $\mathbb{F}_q$, up to pre- and post-composition with independent Möbius transformations. In particular, we obtain a full classification when $q$ is even, and prove an upper bound of $4q$ for the number of equivalence classes when $q$ is odd.
Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the Poisson formula. An alternative approach to Manin's conjecture via universal torsors was used so far mainly over the field Q of rational numbers. In this note, we give a proof of Manin's conjecture over the Gaussian rational numbers Q(i) and over other imaginary quadratic number fields with class number 1 for the singular toric cubic surface defined by t^3=xy…
Let $\mathbb{F}_q$ denote the finite field with $q$ elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over $\mathbb{F}_q$ in terms of the number of rational points on elliptic curves. In the case where $q$ is a prime number, we give a way to calculate these numbers. As a consequence of these results, we characterize maximal and minimal curves given by equations of the forms $ax^3+by^3+cz^3=0$ and $ax^4+by^4+cz^4=0$.
We construct the moduli space, $M_d$, of degree $d$ rational maps on $\mathbb{P}^1$ in terms of invariants of binary forms. We apply this construction to give explicit invariants and equations for $M_3$. Using classical invariant theory, we give solutions to the following problems: (1) explicitly construct, from a moduli point $P\in M_d(k)$, a rational map $ϕ$ with the given moduli; (2) find a model for $ϕ$ over the field of definition (i.e. explicit descent). We work out the method in detail for the cases $d=2,3$.
Let $X$ be a smooth cubic hypersurface of dimension $n \ge 1$ over the rationals. It is well-known that new rational points may be obtained from old ones by secant and tangent constructions. In view of the Mordell--Weil theorem for $n=1$, Manin (1968) asked if there exists a finite set $S$ from which all other rational points can be thus obtained. We give an affirmative answer for $n \ge 48$, showing in fact that we can take the generating set $S$ to consist of just one point. Our proof makes use of a weak approximation theorem due to Skinner, a theorem of Browning, Dietmann and Heath-Brown on…
The Cayley--Salmon theorem implies the existence of a 27-sheeted covering space specifying lines contained in smooth cubic surfaces over $\mathbb{C}$. In this paper we compute the rational cohomology of the total space of this cover, using the spectral sequence in the method of simplicial resolution developed by Vassiliev. The covering map is an isomorphism in cohomology (in fact of mixed Hodge structures) and the cohomology ring is isomorphic to that of $PGL(4,\mathbb{C})$. We derive as a consequence of our theorem that over the finite field $\mathbb{F}_q$ the average number of lines on a cub…
Counting Plane Cubic Curves over Finite Fields with a Prescribed Number of Rational Intersection Points
研究有限域上无公共分量且有规定数量有理交点的平面三次曲线数量，通过设置为射影里德 - 穆勒码的重量枚举器问题，利用计算有公共分量曲线对等方法及麦克威廉姆斯定理变体来求解。
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