Seungki Kim
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.
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In this paper, we present an alternative proof of a finiteness theorem due to Bombieri, Masser and Zannier concerning intersections of a curve in the multiplicative group of dimension n with algebraic subgroups of dimension n-2. The proof uses a method introduced for the first time by Pila and Zannier to give an alternative proof of Manin-Mumford conjecture and a theorem to count points that satisfy a certain number of linear conditions with rational coefficients. This method has been largely used in many different problems in the context of "unlikely intersections".
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.
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。
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 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.
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.
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 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 $(ξ-η)δQ^2$.
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 $Q$.
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 $F(x_1,...,x_n)$ 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 $n$.
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 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 …
A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space P^{n-1}. In this paper, we achieve Serre's conjecture in the special case of smooth cyclic covers of any degree when n is at least 10, and surpass it for covers of degree 3 or higher when n > 10. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsingular leading form, obtained via a combination of an r-th power sieve and the q-analogue of van der Corput's method.
Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over $\mathbb{Q}$ that contains a conic defined over $\mathbb{Q}$.
Counting rational points near manifolds: a refined estimate, a conjecture and a variant
本文改进了接近子流形的有理点计数界，并提出一个猜想，通过复结构重新表述曲率条件，进一步用高斯有理数验证该猜想。
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。
Lehmer-Type bounds and counting rational points of bounded heights on Abelian varieties
本文研究了数域K上阿贝尔簇A的$\bar{K}$-点Néron-Tate高度的Lehmer型界，估计了高度$\leq \log B$的K-有理点数量，并针对椭圆曲线及其乘积将估计中的常数C变为显式可计算。
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.
Cusp Volumes and Rational Points on Spheres
本文证明了Kelmer和Yu关于球面上有理点计数问题中光锥Eisenstein级数Fourier系数的猜想，即每个尖点体积都是有理数。
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 consider the problem of counting the number of rational points on the family of Kummer surfaces associated with two non-isogenous elliptic curves. For this two-parameter family we prove Manin's unity, using the presentation of the Kummer surfaces as isotrivial elliptic fibration and as double cover of the modular elliptic surface of level two. By carrying out the rational point-count with respect to either of the two elliptic fibrations explicitly, we obtain an interesting new identity between two-parameter counting functions.
Counting rationals and diophantine approximation in missing-digit Cantor sets
本文通过傅里叶ℓ^1维度方法，研究了缺失数字康托集中的有理数计数和丢番图逼近问题，并对多个猜想做出了进展。
Counting Rational Points on the Stacky $\operatorname{Sym}^2\mathbb{P}^1$
本文针对商堆叠 $\operatorname{Sym}^2\mathbb{P}^1$ 证明广义 Batyrev–Manin–Malle 猜想的弱形式，并确认其对角为累积子堆叠。
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 paper, we prove a general result computing the number of rational points of bounded height on a projective variety $V$ which is covered by lines. The main technical result used to achieve this is an upper bound on the number of rational points of bounded height on a line. This upper bound is such that it can be easily controlled as the line varies, and hence is used to sum the counting functions of the lines which cover the original variety $V$.
The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of Small Quantum Cohomology of complex Grassmannians is studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann Hypothesis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequenc…
Let $Z$ be a projective geometrically integral algebraic variety. This paper is concerned with estimating the number of rational points on $Z$ which have height at most $B$. The bounds obtained are uniform in varieties of fixed degree and fixed dimension, and are essentially best possible for varieties of degree at least six.
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 say that a subset of $\mathbb{P}^n(\mathbb{R})$ is maximally singular if its contains points with $\mathbb{Q}$-linearly independent homogenous coordinates whose uniform exponent of simultaneous rational approximation is equal to $1$, the maximal possible value. In this paper, we give a criterion which provides many such sets including Grassmannians. We also recover a result of the author and Roy about a class of quadratic hypersurfaces.
We count by height the number of elliptic curves over the rationals that possess an isogeny of degree three.
We count the number of rational elliptic curves of bounded naive height that have a rational $N$-isogeny, for $N \in \{2,3,4,5,6,8,9,12,16,18\}$. For some $N$, this is done by generalizing a method of Harron and Snowden. For the remaining cases, we use the framework of Ellenberg, Satriano and Zureick-Brown, in which the naive height of an elliptic curve is the height of the corresponding point on a moduli stack.
Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a morphism between weighted projective spaces, allowing us to count rational points whose image under this morphism has bounded height. This method provides a more general and simpler proof for a result of the first-named author and Najman on counting elliptic curves with prescribed level structures over number fields. We further include some examples of applicati…
We count flags of primitive lattices, which are objects of the form ${0}=Λ^{(0)}<Λ^{(1)}< \cdots <Λ^{(\ell)}= \mathbb{Z}^n$, where every $Λ^{(i)}$ is a primitive lattice in $\mathbb{Z}^n$. The counting is with respect to two different natural height functions, allowing us to give a new proof of the Manin conjecture for flag varieties over rational numbers. We deduce the equidistribution of rational points in flag varieties, as well as the equidistribution of the shapes of the successive quotient lattices, $Λ^{(i)}/Λ^{(i-1)}$. In doing so, we generalize previous work of Schmidt, as well as our …
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.
Counting rational approximations on rank one flag varieties
该研究针对秩一广义旗簇上随机实点的有理逼近计数问题，运用格空间指数混合与数几何工具，将结果推广至格拉斯曼簇和二次超曲面。
We establish an explicit asymptotic formula for the number of rational solutions of intrinsic Diophantine inequalities on simply-connected simple algebraic groups, at arbitrarily small scales.
We consider a problem posed by Shparlinski, of giving nontrivial bounds for rational exponential sums over the arithmetic function $τ(n)$, counting the number of divisors of $n$. This is done using some ideas of Sathe concerning the distribution in residue classes of the function $ω(n)$, counting the number of prime factors of $n$, to bring the problem into a form where, for general modulus, we may apply a bound of Bourgain concerning exponential sums over subgroups of finite abelian groups and for prime modulus some results of Korobov and Shkredov.
Counting rational points on transcendental curves in valued fields
本文在任意1-h-极小域中建立超越曲线上有理点数的上界，将Pila–Wilkie型计数定理从p-进域推广到混合特征赋值域，方法上避免r次幂映射并采用参数化与行列式方法。
Counting rational maps on $\mathbb{P}^1$ with prescribed local conditions
本文研究了具有指定局部条件的有理映射计数问题，证明了具有最小余数的映射集合具有正密度，并对度数为2的映射进行了显式计算。
In 1999, Neil Calkin and Herbert Wilf wrote "Recounting the rationals" which gave an explicit bijection between the positive integers and the positive rationals. We find several different (some new) ways to construct this enumeration and thus create pointers for generalizing. Next, we use circle packings to generalize and find two other enumerations. Surprisingly, the three enumerations are all that are possible by using this technique. The proofs involve, among other things, "negative" continued fractions, Chebyshev polynomials, Euler's totient function, and generalizations of Stern's diatomi…
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.
Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. We study the asymptotics of the number of geodesics in M starting from and returning to a given cusp, and of the number of horoballs at parabolic fixed points in the universal cover of M. In the appendix, due to K. Belabas, the case of SL(2,Z) and of Bianchi groups is developed.
Counting rational points on Hirzebruch-Kleinschmidt varieties over number fields
本文研究了数域上 Picard 秩为 2 的光滑射影分裂环面簇在 Arakelov 高度下的有理点渐近增长，证明其可分解为有限个不相交子簇并给出显式渐近公式，同时提供了包含 Hirzebruch 曲面的实例。
We relate the problem of counting number fields, in particular, Malle's conjecture with the problem of counting rational points on singular Fano varieties, in particular, Batyrev and Tschinkel's generalization of Manin's conjecture.
For any algebraic variety $V$ defined over a number field $k$, and ample height function $H$ on $V$, one can define the counting function $N_V(B) = #{P\in V(k) \mid H(P)\leq B}$. In this paper, we calculate the counting function for Kummer surfaces $V$ whose associated abelian surface is the product of elliptic curves. In particular, we effectively construct a finite union $C = \cup C_i$ of curves $C_i$ on $V$ such that $N_{V-C}(B)\ll N_C(B)$; that is, $C$ is an accumulating subset of $V$. In the terminology of Batyrev and Manin, this amounts to proving that $C$ is the first layer of the arith…
We study the connection between the singularities of a finite type $\mathbb{Z}$-scheme X and the asymptotic point count of X over various finite rings. In particular, if the generic fiber $X_{\mathbb{Q}}=X\times_{\mathrm{Spec}\mathbb{Z}}\mathrm{Spec}\mathbb{Q}$ is a local complete intersection, we show that the boundedness of $\frac{\left|X(\mathbb{Z}/p^{n}\mathbb{Z})\right|}{p^{n\mathrm{dim}X_{\mathbb{Q}}}}$ in p and n is in fact equivalent to the condition that $X_{\mathbb{Q}}$ is reduced and has rational singularities. This paper completes a result of Aizenbud and Avni.
We consider the problem of counting the number of varieties in a family over a number field which contain a rational point. In particular, for products of Brauer-Severi varieties and closely related counting functions associated to Brauer group elements. Using harmonic analysis on toric varieties, we provide a positive answer to a question of Serre on such counting functions in some cases. We also formulate some conjectures on generalisations of Serre's problem.
We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension $2$. We prove a Mertens counting formula for the rational points over a definite quaternion algebra $A$ over $\mathbb Q$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over $A$ in quaternionic Heisenberg groups.
Counting rational points on Hirzebruch-Kleinschmidt varieties over global function fields
受Bourqui工作启发，本文研究了全局函数域上Picard秩为2的分裂环面簇的高度zeta函数，将其分解为子簇并给出精确解析性质，进而得到各子簇上大高度有理点计数的渐近公式。
Motivic counting of rational curves with tangency conditions via universal torsors
本文借助 Cox 环与泛 torsor 分解曲线到 Mori Dream Space 的态射模空间动机，并在环面簇情形证明满足边界相切条件的 Campana 曲线的动机版 Batyrev–Manin–Peyre 原理。
Counting rational points on elliptic and hyperelliptic curves over function fields
结合2-下降、Riemann-Roch和Bézout定理，给出函数域上椭圆与超椭圆曲线有界高度有理点及S-整点数的上界，并应用于小有限域上雅可比簇挠部分，改进了已知上界。
We consider the problem of counting the number of varieties in a family over $\mathbb{Q}$ with a rational point. We obtain lower bounds for this counting problem for some families over $\mathbb{P}^1$, even if the Hasse principle fails. We also obtain sharp results for some multinorm equations and for specialisations of certain Brauer group elements on higher dimensional projective spaces, where we answer some cases of a question of Serre. Our techniques come from arithmetic geometry and additive combinatorics.
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。
Rational points near planar flat curves
研究平面有限类型曲线附近有理点的渐进行为，推广黄的结果。
Narain lattices are unimodular lattices {\it in} $\R^{r,s}$, subject to certain natural equivalence relation and rationality condition. The problem of describing and counting these rational equivalence classes of Narain lattices in $\R^{2,2}$ has led to an interesting connection to binary forms and their Gauss products, as shown in [HLOYII]. As a sequel, in this paper, we study arbitrary rational Narain lattices and generalize some of our earlier results. In particular in the case of $\R^{2,2}$, a new interpretation of the Gauss product of binary forms brings new light to a number of related o…
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