T. D. Browning, D. R. Heath-Brown
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.
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We investigate the Hasse principle for complete intersections cut out by a quadric and cubic hypersurface defined over the rational numbers.
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.
We study the family of rational curves on arbitrary smooth hypersurfaces of low degree using tools from analytic number theory.
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.
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.
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.
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。
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.
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.
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。
Let $X_4\subset\mathbb{P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field $k$. We show that if either $X_4$ contains a linear subspace $Λ$ of dimension $h\geq \max\{2,\dim(Λ\cap \text{Sing}(X_4))-2\}$ or has double points along a linear subspace of dimension $h\geq 3$, a smooth $k$-rational point and is otherwise general, then $X_4$ is unirational over $k$. This improves previous results by A. Predonzan and J. Harris, B. Mazur, R. Pandharipande for quartics. We also provide a density result for the $k$-rational points of quartic $3$-folds with a double plane over…
Let $X \subset \mathbb{P}^n$ be a non-singular hypersurface of degree $d>1$, and let $ε>0$. This paper is concerned with the conjecture that there are $O(B^{n-1+ε})$ rational points on $X$ that have height at most $B$, in which the implied constant is allowed to depend only upon $d, ε$ and $n$. In particular this conjecture is shown to hold as soon as $d>4$. Furthermore, the main ideas in the proof are used to obtain new paucity estimates for certain diophantine equations.
Slope-determinant method, complex cellular structures and hypersurface coverings of regular rational points
本文结合 Bombieri-Pila/Heath-Brown 行列式方法、Chen 的 Arakelov 斜率重构与 Binyamini-Novikov 复细胞结构，将正则有理点超曲面覆盖计数中的次多项式因子改进为多对数因子。
We survey a number of results on the counting of points on hypersurfaces defined over finite fields. We also investigate when one can be guaranteed a non-singular point on a projective hypersurface and give a condition on the cardinality of the ambient field to achieve this.
On the rationality problem for low degree hypersurfaces
本文研究了低次超曲面的有理性问题，证明了在特定条件下超曲面不具有对角分解，并改进了之前的相关结论。
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.
Quasi algebraically closed fields, or $C_1$ fields, are defined in terms of a low degree condition. Namely, the field $K$ is $C_1$ if every degree $d$ hypersurface of the projective space $\mathbb{P}_K^n$ contains a $K$-point as soon as $d\leq n$. In this article we define a notion of low toric degree generalizing this condition for hypersurfaces of simplicial projective split toric varieties. This allows us to prove a particular case of the $C_1$ conjecture of Kollár, Lang and Manin : any smooth separably rationally connected variety that can be embedded as such a hypersurface over a $C_1$ fi…
Given a smooth projective variety of dimension $n-1\geq 1$ defined over a perfect field $k$ that admits a non-singular hypersurface modelin $\mathbb{P}^n_{\overline{k}}$ over $\overline{k}$, a fixed algebraic closure of $k$, it does not necessarily have a non-singular hypersurface model defined over the base field $k$. We first show an example of such phenomenon: a variety defined over $k$ admitting non-singular hypersurface models but none defined over $k$. We also determine under which conditions a non-singular hypersurface model over $k$ may exist. Now, even assuming that such a smooth hype…
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.
Given a hypersurface, $X$, prime $p$, the zeta function is a generating function for the number of $\mathbb{F}_{p}$ rational points of $X$. Until now, there is no algorithm for computing hypersurfaces with ADE singularities. Scott Stetson and Vladimir Baranovsky provided an algorithm with Mathematica for the ordinary double point case. In this paper, I go over a Sage algorithm for computing the zeta function of a hypersurface with ADE singularities over 3-dimensional projective space. To make the algorithm more efficient, I established an equivalence between a polynomial belonging to the Jacob…
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.
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 $n$.
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。
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 $m_2\geq 5$.
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 $\{ (…
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.
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 $Y$.
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。
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.
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…
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。
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。
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 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。
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 consider the question of how well points in a quadric hypersurface $M\subset\mathbb R^d$ can be approximated by rational points of $\mathbb Q^d\cap M$. This contrasts with the more common setup of approximating points in a manifold by all rational points in $\mathbb Q^d$. We provide complete answers to major questions of Diophantine approximation in this context. Of particular interest are the impact of the real and rational ranks of the defining quadratic form, quantities whose roles in Diophantine approximation have never been previously elucidated. Our methods include a correspondence be…
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