T. D. Browning, D. R. Heath-Brown
Let $F(x_1,...,x_n)$ be a form of degree $d\geq 2$, which produces a geometrically irreducible hypersurface in $\mathbb{P}^{n-1}$. This paper is concerned with the number of rational points on F=0 which have height at most $B$. Whenever $n<6$, or whenever the hypersurface is not a union of lines, we obtain estimates that are essentially best possible and that are uniform in $d$ and $n$.
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An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses the Hardy--Littlewood circle method.
We express the number of points on the Dwork hypersurface $$X_λ^d: x_1^d+x_2^d+\cdots +x_d^d=dλx_1x_2\cdots x_d$$ over a finite field of order $q \not \equiv 1 \pmod{d}$ in terms of McCarthy's $p$-adic hypergeometric function for any odd prime $d$.
We investigate in a statistical fashion the smallest height of a rational point on a Fano hypersurface defined over the field of rational numbers. Along the way, we establish an average version of Manin's conjecture about the number of rational points of bounded height on Fano varieties for the complete family of Fano hypersurfaces of fixed degree and dimension.
We prove asymptotics for the proportion of fibres with a rational point in a conic bundle fibration. The basis of the fibration is a general hypersurface of low degree.
Counting integral points of affine hypersurfaces
该研究针对仿射超曲面上有界高度的整点计数问题，推导得到了一致上界，推广了Browning、Heath-Brown及作者此前的相关研究结果。
We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to this counting function will be given in terms of the degree of the hypersurface, the dimension of the singular locus, the upper bounds of height, and the degree of the field of definition.
Counting rational points on smooth hypersurfaces with high degree
本文研究有理数域上光滑高次超曲面的有界高度有理点计数，给出新上界，并在维数与次数条件下改进维数增长猜想界，同时推广到射影闭包光滑的仿射情形。
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.
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.
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。
Counting rational points on affine hypersurfaces
该研究通过量化Hilbert不可约定理，给出了不可约仿射超曲面上有理点有界高度数目的上界，为数论中的点计数问题提供了新的理论工具。
Let $f(x)=x^5+ax^3+bx^2+cx \in \Z[x]$ and consider the hypersurface of degree five given by the equation \cal{V}_{f}: f(p)+f(q)=f(r)+f(s). Under the assumption $b\neq 0$ we show that there exists $\Q$-unirational elliptic surface contained in $\cal{V}_{f}$. If $b=0, a<0$ and $-a\not\equiv 2,18,34 \pmod {48}$ then there exists $\Q$-rational surface contained in $\cal{V}_{f}$. Moreover, we prove that for each $f$ of degree five there exists $\Q(i)$-rational surface contained in $\cal{V}_{f}$.
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。
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 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.
In this article, we establish an analogue of the dimension growth conjecture, which is regarding the density of rational points on projective varieties, for compact submanifolds of $\mathbb{R}^n$ with non-vanishing curvature. We also establish the convergence theory for the set of simultaneously $ψ$-approximable points lying on a generic hypersurface, thereby settling the generalized Baker-Schmidt problem in the simultaneous setting for generic hypersurfaces. These results are obtained as consequences of an optimal upper bound for the density of rational points near manifolds of the form $\{ (…
Let $F$ be a non-singular homogeneous polynomial of degree $d$ in $n$ variables. We give an asymptotic formula of the pairs of integer points $(\mathbf x, \mathbf y)$ with $|\mathbf x| \le X$ and $|\mathbf y| \le Y$ which generate a line lying in the hypersurface defined by $F$, provided that $n > 2^{d-1}d^4(d+1)(d+2)$. In particular, by restricting to Zariski-open subsets we are able to avoid imposing any conditions on the relative sizes of $X$ and $Y$.
We study the number of representations of an integer n=F(x_1,...,x_s) by a homogeneous form in sufficiently many variables. This is a classical problem in number theory to which the circle method has been succesfully applied to give an asymptotic for the number of such representations where the integer vector (x_1,...,x_s) is restricted to a box of side length P for P sufficiently large. In the special case of Waring's problem, Vaughan and Wooley have recently established for the first time a higher order expansion for the corresponding asymptotic formula. Via a different and much more general…
We provide an asymptotic estimate for the number of rational points of bounded height on a non-singular conic over the rationals. The estimate is uniform in the coefficients of the underlying quadratic form.
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。
We investigate the Hasse principle for complete intersections cut out by a quadric and cubic hypersurface defined over the rational numbers.
We present various improvements to the deformation method for computing the zeta function of smooth projective hypersurfaces over finite fields using $p$-adic cohomology. This includes new bounds for the $p$-adic and $t$-adic precisions required to obtain provably correct results and gains in the efficiency of the individual steps of the method. The algorithm that we thus obtain has lower time and space complexities than existing methods. Moreover, our implementation is more practical and can be applied more generally, which we illustrate with examples of quintic curves and quartic surfaces.
Motivated by a recent question of Peyre, we apply the Hardy-Littlewood circle method to count "sufficiently free" rational points of bounded height on arbitrary smooth projective hypersurfaces of low degree that are defined over the rational numbers.
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。
We prove an estimate on the number of rational points on the Grassmannian variety of bounded twisted height, refining the classical results of Schmidt ([12]) and Thunder ([20]) over the rational field: most importantly, our formula counts all points. Among the consequences are a couple of new implications on the classical subject of counting rational points on flag varieties.
We study the family of rational curves on arbitrary smooth hypersurfaces of low degree using tools from analytic number theory.
Counting elliptic curves over the rationals with a 7-isogeny
本文按高度计数了有理数域上具有7次循环同源的椭圆曲线数量，分别考虑了有理数域及其代数闭包上的同构类。
Counting rational points on the sphere with bounded denominator
本文研究 n 维球面上分母有界的有理点计数问题，并对任意 n≥2 证明了计数余项的最优界。
Counting rational points in non-singular curves
本文应用全局行列式法，给出了非奇异曲线中高度有界的有理点数量的统一上界。
Consider an irreducible bilinear form $f(x_1,x_2;y_1,y_2)$ with integer coefficients. We derive an upper bound for the number of integer points $(\mathbf{x},\mathbf{y})\in\mathbb{P}^1\times\mathbb{P}^1$ inside a box satisfying the equation $f=0$. Our bound seems to be the best possible bound and the main term decreases with a larger determinant of the form $f$. We further discuss the case when $f(x_1,x_2;y_1,y_2;z_1,z_2)$ is an irreducible non-singular trilinear form defined on $\mathbb{P}^1\times \mathbb{P}^1\times\mathbb{P}^1$, with integer coefficients. In this case, we examine the singular…
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 non-singular projective hypersurface of degree 4, which is defined over the rational numbers. Assume that X has dimension 39 or more, and that X contains a real point and p-adic points for every prime p. Then X is shown to contain infinitely many rational points.
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.
In this article we present a new method to obtain polynomial lower bounds for Galois orbits of torsion points of one dimensional group varieties.
On the number of rational points of Artin-Schreier curves and hypersurfaces
本文利用二次型与置换矩阵方法，精确计算了有限域上两类Artin-Schreier曲线和超曲面的有理点个数，并证明Weil界仅在参数λ的迹为零时达到。
For any number field k, upper bounds are established for the number of k-rational points of bounded height on non-singular del Pezzo surfaces defined over k, which are equipped with suitable conic bundle structures over k.
Rational lines on diagonal hypersurfaces and subconvexity via the circle method
本文通过圆法，在$s\ge k(k+1)$条件下证明了对角超曲面上整数坐标直线的Hasse原理，突破了平方根抵消的凸性障碍。
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.
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.
Rational surfaces on low degree hypersurfaces
该研究借助函数域解析数论，明确了从射影平面到足够低次光滑超曲面的固定次态射模空间的不可约性与维数。
We find an asymptotic formula for the number of rational points near planar curves. More precisely, if $f:\mathbb{R}\rightarrow\mathbb{R}$ is a sufficiently smooth function defined on the interval $[η,ξ]$, then the number of rational points with denominator no larger than $Q$ that lie within a $δ$-neighborhood of the graph of $f$ is shown to be asymptotically equivalent to $(ξ-η)δQ^2$.
This paper establishes the conjecture that a non-singular projective hypersurface of dimension $r$, which is not equal to a linear space, contains $O(B^{r+ε})$ rational points of height at most $B$, for any choice of $ε>0$. The implied constant in this estimate depends at most upon $ε, r$ and the degree of the hypersurface.
We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for typical forms $Q$.
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.
Counting odd genus $2$ curves with a marked rational $3$-torsion point
研究有理数域上特定条件的亏格2曲线，通过朴素高度排序计数，核心方法是基于曲线的首一魏尔斯特拉斯模型及雅可比行列式的有理3 - 挠点条件。
Let $Z$ be a quadratic hypersurface of $\mathbb{P}^n(\mathbb{R})$ defined over $\mathbb{Q}$ containing points whose coordinates are linearly independent over $\mathbb{Q}$. We show that, among these points, the largest exponent of uniform rational approximation is the inverse $1/ρ$ of an explicit Pisot number $ρ<2$ depending only on $n$ if the Witt index (over $\mathbb{Q}$) of the quadratic form $q$ defining $Z$ is at most $1$, and that it is equal to $1$ otherwise. Furthermore there are points of $Z$ which realize this maximum. They constitute a countably infinite set in the first case, and an…
Distribution of rational points on random Fano hypersurfaces
本文证明了随机 Fano 超曲面上有理点平均数量的渐近公式，以及容许 adelic 点的超曲面数量的渐近公式。
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.
Let $n$ be a positive multiple of $4$. We establish an asymptotic formula for the number of rational points of bounded height on singular cubic hypersurfaces $S_n$ defined by $$ x^3=(y_1^2 + \cdots + y_n^2)z . $$ This result is new in two aspects: first, it can be viewed as a modest start on the study of density of rational points on those singular cubic hypersurfaces which are not covered by the classical theorems of Davenport or Heath-Brown; second, it proves Manin's conjecture for singular cubic hypersurfaces $S_n$ defined above.
Counting modular forms by rationality field
本文研究权为2新形式的有理性域分布，给出无平方因子级下次数d出现频率的启发式上界，预测d≥7时有限，并猜想Q(√5)是最常见二次有理性域。
For any affine hypersurface defined by a complete symmetric polynomial in $k\geq 3$ variables of degree $m$ over the finite field $\mathbb{F}_{q}$ of $q$ elements, a special case of our theorem says that this hypersurface has at least $6q^{k-3}$ rational points over $\mathbb{F}_{q}$ if $1\leq m \leq q-3$ and $q$ is odd. A key ingredient in our proof is Segre's classical theorem on ovals in finite projective planes.
We design and analyze an algorithm for computing rational points of hypersurfaces defined over a finite field based on searches on "vertical strips", namely searches on parallel lines in a given direction. Our results show that, on average, less than two searches suffice to obtain a rational point. We also analyze the probability distribution of outputs, using the notion of Shannon entropy, and prove that the algorithm is somewhat close to any "ideal" equidistributed algorithm.
Let $E/\mathbb{Q}$ be an elliptic curve over the rational numbers. It is known, by the work of Bombieri and Zannier, that if $E$ has full rational $2$-torsion, the number $N_E(B)$ of rational points with Weil height bounded by $B$ is $\exp\left(O\left(\frac{\log B}{\sqrt{\log\log B}}\right)\right)$. In this paper we exploit the method of descent via $2$-isogeny to extend this result to elliptic curves with just one nontrivial rational $2$-torsion point. Moreover, we make use of a result of Petsche to derive the stronger upper bound $N_{E}(B) = \exp\left(O\left(\frac{\log B}{\log\log B}\right)\…
Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to a given $w>0$ is determined. The counting functions developed are framed in terms of convolution exponentials, and are closely related to the Riemann zeta function. At the conclusion, several estimates are derived, with special emphasis on the case of sums of square roots, i.e. $j=1$, $k=2$.
We establish a sharp asymptotic formula for the number of rational points up to a given height and within a given distance from a hypersurface. Our main innovation is a bootstrap method that relies on the synthesis of Poisson summation, projective duality and the method of stationary phase. This has surprising applications to counting rational points lying on the manifold; indeed, we are able to prove an analogue of Serre's Dimension Growth Conjecture (originally stated for projective varieties) in this general setup. As another consequence of our main counting result, we obtain an optimal Jar…
In this paper, we consider a problem of counting multiplicities. We fix a counting function of multiplicity of rational points in a hypersurface of a projective space over a finite field, and we give an upper bound for the sum with respect to this counting function in terms of the degree of the hypersurface, the dimension and the cardinality of the finite field. This upper bound gives a description of the complexity of the singular locus of this hypersurface. In order to obtain this upper bound, we introduce a notion called intersection tree by intersection theory. We construct a sequence of i…
Free rational curves on low degree hypersurfaces and the circle method
本文利用 Hardy–Littlewood 圆法的函数域版本分析低次数光滑射影超曲面上的自由有理曲线，从而界定有理曲线模空间奇异轨迹维数，并解释 Peyre 对 Batyrev–Manin 猜想的斜率化表述。
Lattice points in thickened parabolas and rational points near hypersurfaces
研究厚化抛物线中有理点的分布，利用动力学方法和Ratner定理，改进了超曲面附近有理点逼近的下界结果。
We find upper and lower bounds on the number of rational points that are $ψ$-approximations of some $n$-dimensional $p$-adic integer. Lattice point counting techniques are used to find the upper bound result, and a Pigeon-hole principle style argument is used to find the lower bound result. We use these results to find the Hausdorff dimension for the set of $p$-adic weighted simultaneously approximable points intersected with $p$-adic coordinate hyperplanes. For the lower bound result we show that the set of rational points that $τ$-approximate a $p$-adic integer form a set of resonant points …
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