V. Vinay Kumaraswamy
Let $F \in \mathbf{Z}[\boldsymbol{x}]$ be a diagonal, non-singular quadratic form in $4$ variables. Let $λ(n)$ be the normalised Fourier coefficients of a holomorphic Hecke form of full level. We give an upper bound for the problem of counting integer zeros of $F$ with $|\boldsymbol{x}| \leq X$, weighted by $λ(x_1)$.
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On the arithmetic of intersections of two quadrics containing a conic
本文重录1993年未发表手稿，证明了Colliot-Thélène、Sansuc与Swinnerton-Dyer关于含圆锥曲线的两个二次曲面交算术性质的两个猜想。
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 $Q$.
We establish the Hardy-Littlewood property (à la Borovoi-Rudnick) for Zariski open subsets in affine quadrics of the form $q(x_1,\cdots,x_n)=m$, where $q$ is a non-degenerate integral quadratic form in $n\geqslant 3$ variables and $m$ is a non-zero integer. This gives asymptotic formulas for the density of integral points taking coprime polynomial values, which is a quantitative version of the arithmetic purity of strong approximation property off infinity for affine quadrics.
We construct a torsion-free arithmetic lattice in $\mathrm{PGL}_2(\mathbb{F}_2(\!(t)\!))\times\mathrm{PGL}_2(\mathbb{F}_2(\!(t)\!))$ arising from a quaternion algebra over $\mathbb{F}_2(z)$. It is the fundamental group of a square complex with universal covering $T_3\times T_3$, a product of trees with constant valency $3$, which has minimal Euler characteristic. Furthermore, our lattice gives rise to a fake quadric over $\mathbb{F}_2(\!(t)\!)$ by means of non-archimedean uniformization.
Effective count of integer points on ternary affine quadrics and effective equidistribution
通过混合性研究数字段上特定无限均匀测度的等分布及计数问题，获得Oh-Shah和后续Kelmer-Kontorovich研究的平滑版本。
Effective equidistribution, arithmetic purity of strong approximation, and geometric sieve for affine quadrics
本文利用对称空间中 $S$-整点的有效等分布，证明仿射二次曲面在去掉任意单个位置时强逼近的算术纯粹性，并在有理数域上建立 $p_0$-整点的几何筛。
A fake quadric is a smooth projective surface that has the same rational cohomology as a smooth quadric surface but is not biholomorphic to one. We provide an explicit classification of all irreducible fake quadrics according to the commensurability class of their fundamental group. To accomplish this task, we develop a number of new techniques that explicitly bound the arithmetic invariants of a fake quadric and more generally of an arithmetic manifold of bounded volume arising from a form of SL_2 over a number field.
A pair of symmetric bilinear forms A and B determine a binary form $f(x,y) = disc(Ax-By)$. We prove that the question of whether a given binary form can be written in this way as a discriminant form generically satisfies a local-global principle and deduce from this that most binary forms over $\mathbf{Q}$ are discriminant forms. This is related to the arithmetic of the hyperelliptic curve $z^2 = f(x,y)$. Analogous results for non-hyperelliptic curves are also given.
Quadrics in arithmetic statistics
本文将圆法与 Bhargava 计数技术结合，提出一种处理由二次不变量定义的仿射空间子簇轨道计数问题的通用方法，并通过计算 2-Selmer 群平均大小验证其有效性。
We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics without an explicit error term, we provide an explicit error term by combining their work with the method of Hooley regarding Artin's primitive root conjecture. We give an application to a Diophantine problem involving primes with a given primitive root.
We prove a Diophantine approximation inequality for rational points in varieties of any dimension, in the direction of Vojta's conjecture with truncated counting functions. Our results also provide a bound towards the $abc$ conjecture which in several cases is subexponential. The main theorem gives a lower bound for the truncated counting function relative to a divisor with sufficiently many components, in terms of the proximity to an algebraic point. Furthermore, we show that the Lang-Waldschmidt conjecture implies a special case of Vojta's conjecture with truncation in arbitrary dimension. O…
On arithmetic terms expressing the prime-counting function and the n-th prime
本研究提出首个固定长度的初等闭式算术项，可分别表示素数计数函数$\pi(n)$与第$n$个素数$p(n)$，仅用有限固定次数的初等算术运算，为素数是否存在次序的基础问题提供构造性解答。
Counting integer points on affine surfaces with a side condition
本研究基于行列式法拓展前人工作，给出带多项式同余条件的有界高度仿射曲面整数点的一致上界，并将其应用于对角二次曲面点计数及四异幂和整数表示问题。
Let $X(D,1) =Γ(D,1) \backslash \mathbb{H}$ denote the Shimura curve of level $N=1$ arising from an indefinite quaternion algebra of fixed discriminant $D$. We study the discrete average of the error term in the hyperbolic circle problem over Heegner points of discriminant $d <0$ on $X(D,1)$ as $d \to -\infty$. We prove that if $|d|$ is sufficiently large compared to the radius $r \approx \log X$ of the circle, we can improve on the classical $O(X^{2/3})$-bound of Selberg. Our result extends the result of Petridis and Risager for the modular surface to arithmetic compact Riemann surfaces.
Counting the number of integral fixed points of a discrete dynamical system with applications from arithmetic statistics, I
本文研究了离散动力系统整数不动点的计数问题，通过算术统计方法得出了关于不同素数下不动点数量的统计结果。
Counting maximal arithmetic subgroups
本文推广Borel和Prasad的方法，研究了半单李群中有界余体积的极大算术子群个数的增长率.
A non-flag arithmetic regularity lemma and counting lemma
本文提出了一个不依赖旗形条件的算术正则引理和计数引理，适用于所有线性形式系统，推广了Green和Tao的经典结果。
For any $k>1$, we find the asymptotics of the counting function of $k$-th power-free elements in an additive arithmetic semigroup with exponential growth of the abstract prime counting function. This paper continues the authors' earlier research dealing with the case of infinite $k$.
On smooth elements in arithmetic semigroups
本文研究算术半群中的光滑元素，在满足公理A时给出计数函数的显式渐近式，推广了整数情形结果，并应用于有限阿贝尔群和半单环的光滑族计数。
On the arithmetic of crossratios and generalised Mertens' formulas
我们发展双曲几何和算术equidistribution问题之间的关系，所产生的行动的算术群对真正的双曲空间，特别是在维高达5。我们证明了二次虚数域和有理数上的定四元数代数的Mertens公式的推广，关于两种不同自然复杂度的二次无理数的计数结果，以及具有误差界的二进制二次、Hermitian和Hamilton形式的（代数）整数表示的计数结果。对于每一个这样的陈述，我们证明了相应的算术定义点的等分布结果。进一步，我们研究了这类点的交比的渐近性质，推广了Pollicott关于Schottky-Klein素函数的最新结果.
Solvable points on intersections of quadrics, cubics, and quartics
本文建立多项式下界，使得在二次、三次和四次超曲面交线X上存在稠密的可解点集，方法是将极超曲面经典理论与X上j维线性子空间的Fano簇联系起来。
We relate the singularities of a scheme $X$ to the asymptotics of the number of points of $X$ over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if $Γ$ is an arithmetic lattice whose $\mathbb{Q}$-rank is greater than one, let $r_n(Γ)$ be the number of irreducible $n$-dimensional representations of $Γ$ up to isomorphism. We prove that there is a constant $C$ (for example, $C=746$ suffices) such that $r_n(Γ)=O(n^C)$ for every such $Γ$. This answers a question of Larsen and Lubotzky.
An arithmetic formula is an expression involving only the constant $1$, and the binary operations of addition and multiplication, with multiplication by $1$ not allowed. We obtain an asymptotic formula for the number of arithmetic formulas evaluating to $n$ as $n$ goes to infinity, solving a conjecture of E. K. Gnang and D. Zeilberger. We give also an asymptotic formula for the number of arithmetic formulas evaluating to $n$ and using exactly $k$ multiplications. Finally we analyze three specific encodings for producing arithmetic formulas. For almost all integers $n$, we compare the lengths o…
This paper, motivated by problems in Diophantine analysis which can be formulated as problems of finding rational points on the intersection of two quadrics, presents an explicit construction of a rationally defined isomorphism (biregular mapping) between a rationally defined smooth intersection of two quadrics in projective three-space and an elliptic curve in Weierstrass form which maps a distinguished rational point to the point at infinity. The usual approach of transforming a smooth plane cubic to a curve in Weierstrass form by mapping an inflection point to the point at infinity in a par…
Generalised quadratic forms over totally real number fields
本文引入全实数域上一类新的广义二次型，并通过数域上的Hardy-Littlewood圆法证明其能刻画有理数上任意二次曲面组的算术性质。
A method is described to sum multi-dimensional arithmetic functions subject to hyperbolic summation conditions, provided that asymptotic formulae in rectangular boxes are available. In combination with the circle method, the new method is a versatile tool to count rational points on algebraic varieties defined by multi-homogeneous diagonal equations.
Counting the number of inequivalent arithmetic expressions on $n$ variables
本文提出了一种时间复杂度为Θ(n²)的算法，用于计算n个不同变量上不等价的算术表达式的数量。
We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary number fields $K$ in the light cone of Hermitian forms, as well as an equidistribution theorem of the set of rational points over $K$ in Heisenberg groups. We give a counting formula for the cubic points over $K$ in the complex projective plane whose Galois conjugates are orthogonal…
Counting the number of $n$-periodic integral points of a discrete dynamical system with applications from arithmetic statistics, IV
本文研究了离散动力系统中n-周期整数点的计数问题，揭示了系数c与n-周期点数量的关系，并应用算术统计中的方法获得多项式、数域和Artin L-函数的统计结果。
A sieve for rational points on suitable varieties is developed, together with applications to counting rational points in thin sets, the number of varieties in a family which are everywhere locally soluble, and to the notion of friable rational points with respect to divisors. In the special case of quadrics, sharper estimates are obtained by developing a version of the Selberg sieve for rational points.
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 the number of $m$-periodic $\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, V
本文研究了多项式映射中m-周期点的计数问题，揭示了系数c与周期m之间的关系，并通过算术统计方法得出了关于代数数域和ArtinL-函数的统计结果。
Counting arithmetic lattices and surfaces
给出了单李群$H$中余体积至多为$x$的算术格$Γ$的个数$AL_H(x)$的估计.特别地，我们得到了体积至多为$x$的算术3-流形个数的第一个具体估计。我们的主要结果是经典的情况下$H=PSL(2,R)$，我们计算的极限时，$x\to\infty$的$\log AL_H(x) / x\log x$。证明使用几种不同的技术：几何（界定的数量生成器的$Γ$作为其covume的函数），数论（界定的数量最大的这样的$Γ$）和尖锐的估计的特征值的对称群（界定的子群增长的$Γ$）。
Short Recurrences and Straight-Line Programs for Primes
该研究提出从初始值8生成奇素数的短自主递推式，将素数计数函数表示为整数的2-adic赋值，还构造出对应素数相关函数的固定算术直线程序。
Given a symmetric variety Y defined over the rationals and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski dense set of integral points on Y. We also establish an asymptotic counting formula for this set. In the special case that Y is a quadric hypersurface, we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics.
It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic $3$-dimensional orbifold defines $c Q^{1/2} + O(Q^{1/4})$ square-rootable Salem numbers of degree $4$ which are less than or equal to $Q$. This quantity can be compared to the total number of such Salem numbers, which is shown to be asymptotic to $\frac{4}{3}Q^{3/2}+O(Q)$. Assuming the gap conjecture of Marklof, we can extend these results to compact arithmetic $3$-orbifolds. As an applic…
Counting integral points on symmetric varieties with applications to arithmetic statistics
本文结合 Bhargava 几何数论与动力点计数方法，提出对称簇整点计数新技术，并应用于三次数域类群 2-挠子群和椭圆曲线 2-Selmer 群的薄族分布，揭示了小范数生成元对平均大小的增大效应。
Counting the number of $1_{m}$-preperiodic $\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, VII
本文研究多项式映射 $\varphi_{d,c}(z)=z^d+c$ 的 $1_n$-预周期整点与系数 $c$ 的关系，证明对奇次和偶次映射，模 $p$ 的不同 $1_n$-预周期整点平均数量随 $c$ 增长无界或为零，并应用算术统计中的密度、计数和等分布结果。
Counting the number of $n$-periodic $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VI
本文研究了多项式映射φ_{d,c}的n-周期点数量，揭示了其与系数c的关系，并在算术统计中应用了密度和等分布结果。
Counting the number of $\mathcal{O}_{K}$-fixed points of a discrete dynamical system with applications from arithmetic statistics, II
本文研究了离散动力系统中 $\mathcal{O}_{K}$-不变点的计数问题，并在算术统计中应用了相关结果。
Counting Salem numbers arising from arithmetic hyperbolic orbifolds
本文研究算术双曲轨道中Salem数与短测地线的关系，通过界定了Salem数比例和测地线谱乘法的指数增长下界，改进了相关理论。
Counting appearances of integers in sets of arithmetic progressions
通过行列式方法，研究了整数在由k个不同素数生成的等差数列余数类中的出现次数计数问题。
In this paper we introduce and study a family $Φ_k$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equation $\gcd(x_1^2+\ldots +x_k^2, n)=1$ with $x_1,\ldots,x_k \in {\mathbb{Z}}/n{\mathbb{Z}}$ which, for $k=2,4$ and $8$ coincide, respectively, with the number of units in the rings of Gaussian integers, quaternions and octonions over ${\mathbb{Z}}/n{\mathbb{Z}}$. We prove that $Φ_k$ is multiplicative for every $k$, we obtain an explicit formula for $Φ_k(n)$ in terms of the prime-power decomposition of $n$ and derive an…
Counting polynomials with positive roots
本文研究所有根均为正实数的首一整系数多项式的计数问题，通过估计满足麦克劳林不等式的整数序列建立固定迹的渐近公式，并对三次多项式给出更精确结果，还证明正比例的三次多项式具有无平方因子判别式。
Counting the number of $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-fixed points of a discrete dynamical system with applications from arithmetic statistics, III
本文研究多项式动力系统中$\mathbb{Z}_{p}$-和$\mathbb{F}_{p}[t]$-固定点的计数问题，并应用算术统计结果获得数域和函数场的统计结果。
Diophantine and Non-Diophantine Arithmetics: Operations with Numbers in Science and Everyday Life
科学和数学帮助人们更好地理解世界，消除各种谬论和误解。其中一个误解与算术有关，算术对科学和日常生活都很重要。人们认为他们的计数是由传统算术规则支配的，其他种类的算术不存在，也不可能存在。本文证明，这种流行的形象的情况下，整数是不正确的。在许多情况下，我们必须使用不同的计算和操作规则。这是自然界和社会中存在的多样性的结果，为了正确地表示这种多样性，人们必须使用不同的算法。为了区分它们，我们称常规算术为丢番图，而其他算术称为非丢番图。非丢番图算术的理论是在作者的《非丢番图算术或2 + 2不等于4的可能性》一书中发展起来的。在这项工作中，非丢番图算术的一些性质被认为是，以及它们的连接到数值计算和当代物理学的解释。
Counting Arithmetical Structures on Paths and Cycles
本文研究路径图和环图上的算术结构，证明路径上的算术结构由Catalan数枚举，环上的由二项式系数枚举，并确定了所有算术结构的临界群。
Counting subsets of integers free of arithmetic configurations
本文改进了关于无等差数列子集计数的问题，证明了对于k≥5，无限多个n值下计数结果为2^{r_k(n)(1+o(1))}，并为k≥3的所有n提供了2^{O(r_k(n))}的计数结果。
On the Growth of the Counting Function of Stanley Sequences
给定一个非负整数的有限集A，A不具有3项等差数列，由A生成的Stanley序列，记作S（A），是从A开始，然后包含严格较大的整数，这些整数在S（A）中不引入3项等差数列而生成的无限集。Erdos等人提出了一个问题，即Stanley序列S（A）的计数函数S（A，x）是否满足S（A，x）>x^{1/2-ε}，对于每个ε>0和x>x_0（ε，A）。在本文中，我们肯定地回答了这个问题，事实上，我们证明了一个稍微强一些的结果：S（A，x）\geq（\sqrt{2} -ε）\sqrt{x}对于x \geq x_0（ε，A）.
Equidistribution of integers represented by standard quadratic form under arithmetic constraints
研究算术约束下标准二次型表示整数的等分布，通过构造模形式、利用表示理论确定同余子群并建立尖点形式充要条件，证明了 $d \geq 4$ 时等分布在 $p+1$ 个轨道上局部发生。
We report on our project to find explicit examples of $K3$ surfaces having real or complex multiplication. Our strategy is to search through the arithmetic consequences of RM and CM. In order to do this, an efficient method is needed for point counting on surfaces defined over finite fields. For this, we describe algorithms that are $p$-adic in nature.
In this book, we study equidistribution and counting problems concerning locally geodesic arcs in negatively curved spaces endowed with potentials, and we deduce, from the special case of tree quotients, various arithmetic applications to equidistribution and counting problems in non-Archimedean local fields.
Quantum Algorithms for Gowers Norm Estimation, Polynomial Testing, and Arithmetic Progression Counting over Finite Abelian Groups
本研究提出有限阿贝尔群上Gowers范数估计的量子算法，推广至任意域与高阶k，应用于多项式测试、等差数列计数，且具噪声鲁棒性，可用于NISQ时代量子属性测试等领域。
Arithmetic functions and fixed points of powers of permutations
本文研究置换幂的不动点计数函数如何确定置换的共轭类，构造了计算共轭类的算法，并刻画了可作为不动点计数函数的算术函数。
Counting quadratic points on Fano varieties
本文研究弱 Fano 簇对称平方上的有界高度二次点计数，建立一般框架并对无限族非分裂二次曲面证明 Manin–Peyre 猜想，同时推广整数环格点计数与二次扩张 Euler 乘积工具。
Orbit-counting in non-hyperbolic dynamical systems
对于具有双曲行为的动力系统，有著名的素数定理和Mertens定理的类似物。这里我们考虑最简单的非双曲代数系统的相同问题。轨道计数函数的渐进行为由相关紧群上的旋转决定，并且在简单的例子中，我们展示了轨道计数函数的无数个不同的渐进增长率。梅尔滕斯定理也在这种情况下成立，一个明确的合理的首项系数从算术性质的非双曲特征方向。
Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications
研究了在算术同构空间中格点的大翻译的有效等分布，应用于3×3整矩阵的渐近计数公式，改进了Eskin-Mozes-Shah的经典定理。
In arithmetic statistics and analytic number theory, the asymptotic growth rate of counting functions giving the number of objects with order below $X$ is studied as $X\to \infty$. We define general counting functions which count epimorphisms out of an object on a category under some ordering. Given a probability measure $μ$ on the isomorphism classes of the category with sufficient respect for a product structure, we prove a version of the Law of Large Numbers to give the asymptotic growth rate as $X$ tends towards $\infty$ of such functions with probability $1$ in terms of the finite moments…
Counting Rational Points on K3 Surfaces
对于在数域$k$上定义的任何代数簇$V$，以及$V$上的充分高度函数$H$，可以定义计数函数$N_V(B) = #{P\in V(k) \mid H(P)\leq B}$。本文计算了交换曲面为椭圆曲线乘积的Kummer曲面$V$的计数函数。特别地，我们有效地构造了曲线$C_i$在$V$上的有限并集$C = \cup C_i$，使得$N_{V-C}(B)\ll N_C(B)$，即$C$是$V$的累积子集。在Batyrev和Manin的术语中，这相当于证明$C$是$V$算术分层的第一层。
In [BGLM] and [GLNP] it was conjectured that if $H$ is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in $H$ of covolume at most $x$ is $x^{(γ(H)+o(1))\log x/\log\log x}$ where $γ(H)$ is an explicit constant computable from the (absolute) root system of $H$. In [BLu] we disproved this conjecture. In this paper we prove that for most groups $H$ the conjecture is actually true if we restrict to counting only non-uniform lattices.
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