Brett Nasserden, Stanley Yao Xiao
In this paper we consider the density of rational points on the "stacky" curve $\mathcal{X}(\mathbb{P}^1;0,2;1,2;\infty,2)$ which is $\mathbb{P}^1$ with three half points, with respect to the so-called Ellenberg-Satriano-Zuerick-Brown height. In particular, we prove a conjecture of Ellenberg.
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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。
We construct a stacky curve of genus $1/2$ (i.e., Euler characteristic $1$) over $\mathbb{Z}$ that has an $\mathbb{R}$-point and a $\mathbb{Z}_p$-point for every prime $p$ but no $\mathbb{Z}$-point. This is best possible: we also prove that any stacky curve of genus less than $1/2$ over a ring of $S$-integers of a global field satisfies the local-global principle for integral points.
The stacky Batyrev-Manin conjecture and modular curves
本文研究了在有理数域上，模曲线X₀(N)上的叠层Batyrev-Manin猜想成立的条件，并给出了X₀(N)作为平方根栈覆盖叠层曲线的明确描述。
Heights and quantitative arithmetic on stacky curves
本文研究堆叠曲线上的E-S-ZB高度，构造对偶高度、计数有界高度有理点，并揭示其Northcott性质及Vojta猜想与abc猜想的等价性。
A stacky nilpotent $p$-adic Riemann-Hilbert correspondence
本文利用叠论语言，通过引入 $\mathbb B^+_{dR}$-局部系统与 $t$-连接的模叠，证明了光滑刚体簇上的幂零 $p$-进 Riemann-Hilbert 对应。
A weighted projective stack is a stacky quotient $\mathscr P(\mathbf a)=(\mathbf A^n-\{0\})/\mathbb G_m$, where the action of $\mathbb G_m$ is with weights $\mathbf a\in\mathbb Z^n_{>0}$. Examples are: the compactified moduli stack of elliptic curves $\mathscr P(4,6)$ and the classifying stack of $μ_m$-torsors $Bμ_m=\mathscr P(m)$. We define heights on the weighted projective stacks. The heights generalize the naive height of an elliptic curve and the absolute discriminant of a torsor. We use the heights to count rational points. We find the asymptotic behaviour for the number of rational poin…
We analyze the geometry of rational p-division points in degenerating families of elliptic curves in characteristic p. We classify the possible Kodaira symbols and determine for the Igusa moduli problem the reduction type of the universal curve. Special attention is paid to characteristic 2 and 3 where wild ramification and stacky phenomena show up.
Section Rings of $\mathbb{Q}$-Divisors on Genus $1$ Curves
本文研究椭圆曲线上有理除子的截面环，推广已有亏格零及堆叠曲线相关结果，给出两点支集有效除子的生成元与关系首项，并界定任意有限点支集情形的生成元次数。
Multi-height distribution of rational points of split toric stacks
通过将计数问题提升到与轨道皮卡德群相关的环面扩展泛旋子，研究了所有高度在Q上有界的分裂环状栈的有理点分布，证明了有理点的整数参数化存在性，并定义了环状栈的玉河数。
Stacky Heights, Entropy, and Zeta Functions of Weighted Hypersurfaces over Finite Fields
本文证明加权超曲面的规范堆栈zeta函数具有有限扇区分解，给出特殊值的有理性与函数方程，并建立迷向熵的Lang–Weil渐近，将函数域高度等同于稳定堆栈高度。
A Kloosterman refinement for function fields $K=\mathbb{F}_q(t)$ is developed and used to establish the quantitative arithmetic of the set of rational points on a smooth complete intersection of two quadrics $X\subset \mathbb{P}^{n-1}_{K}$ , under the assumption that $q$ is odd and $n\geq 9$.
We give a list of $PGL_2(\mathbb{F}_\ell)$ number fields for $\ell \geq 11$ which have rational companion forms. Our list has fifty-three fields and seems likely to be complete. Some of the fields on our list are very lightly ramified for their Galois group.
Every finite abelian group is the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$
本文证明每个有限阿贝尔群在特征2、3、5的有限域上可实现为普通阿贝尔簇的有理点群，并给出一般有限域的部分结果及循环群的无穷多实现。
An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface $x_1y_1^2+\dots+x_4y_4^2=0$ in $\mathbb{P}^3\times\mathbb{P}^3$. This confirms the modified Manin conjecture for this variety, in which the removal of a thin set of rational points is allowed.
Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(D_4^\infty)$ be the compositum of all extensions of $\mathbb{Q}$ whose Galois closure has Galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of $D_4$. In this article we first show that $\mathbb{Q}(D_4^\infty)$ is in fact the compositum of all $D_4$ extensions of $\mathbb{Q}$ and then we prove that the torsion subgroup of $E(\mathbb{Q}(D_4^\infty))$ is finite and determine the 24 possibilities for its structure. We also give a complete classification of the elliptic curves that have…
In this paper we mainly study sums of four rational squares with certain restrictions. Let $\mathbb Q_{\ge0}$ be the set of nonnegative rational numbers. We establish the following four-square theorem for rational numbers: For any $a,b,c,d\in\mathbb Q_{\ge0}$, each $r\in\mathbb Q_{\ge0}$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Q_{\ge0}$ such that $ax+by+cz+dw$ is a rational square (or a rational cube). This paper also contains many conjectures; for example, for any positive integers $a$ and $b$ with $\gcd(a,b)=1$, we conjecture that each $r\in\mathbb Q_{\ge0}$ can be writte…
Rational points in a family of conics over $\mathbb{F}_2(t)$
本研究将Serre关于有理数域上平面圆锥曲线有理点的经典结论拓展到全局函数域，聚焦$\u211d_2(t)$上的一族圆锥曲线，借助调和分析和函数域上的Tauberian定理得到渐近公式，揭示了新的数论特性。
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 $\approx 97.3\%$.
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 over $\mathbb{Q}$ such that $\mathrm{End}_{\bar{\mathbb{Q}}}(E)=\mathbb{Z}$ and which admits a non-trivial cyclic $\mathbb{Q}$-isogeny. We prove that, for $p>37$, the residual mod $p$ Galois representation $\barρ_{E,p}:G_{\mathbb{Q}}\rightarrow\mathrm{GL}_2(\mathbb{F}_p)$ is surjective.
Curves with prescribed rational points
针对Faltings定理的反向问题——给定有限有理点集是否存在对应有理点集恰好为该集合的光滑曲线，本文提出有效构造算法并给出肯定答案。
Let $φ:\mathbb{P}^1(\mathbb F_q)\to\mathbb{P}^1(\mathbb F_q)$ be a rational map of degree $d>1$ on a fixed finite field. We give asymptotic formulas for the size of image sets $φ^n(\mathbb{P}^1(\mathbb F_q))$ as a function of $n$. This is done using properties of Galois groups of iterated maps, whose connection to the size of image sets is established via the Chebotarev Density Theorem. We apply our results in the following setting. For a rational map defined over a number field, consider the reduction of the map modulo each prime of the number field. We use our results to give explicit bounds…
Let $N$ be a non-squarefree positive integer and let $\ell$ be an odd prime such that $\ell^2$ does not divide $N$. Consider the Hecke ring $\mathbb{T}(N)$ of weight $2$ for $Γ_0(N)$, and its rational Eisenstein primes of $\mathbb{T}(N)$ containing $\ell$, defined in Section 3. If $\mathfrak{m}$ is such a rational Eisenstein prime, then we prove that $\mathfrak{m}$ is of the form $(\ell, ~\mathcal{I}^D_{M, N})$, where the ideal $\mathcal{I}^D_{M, N}$ of $\mathbb{T}(N)$ is also defined in Section 3. Furthermore, we prove that $\mathcal{C}(N)[\mathfrak{m}] \neq 0$, where $\mathcal{C}(N)$ is the …
In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/k\mathbb{Z}$ for $k = 2, 4, 6, 8$, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.
Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $\mathbb{Q}^{ab}$ be the maximal abelian extension of $\mathbb{Q}$. In this article we classify the groups that can arise as $E(\mathbb{Q}^{ab})_{\text{tors}}$ up to isomorphism. The method illustrates techniques for finding explicit models of modular curves of mixed level structure. Moreover we provide an explicit algorithm to compute $E(\mathbb{Q}^{ab})_{\text{tors}}$ for any elliptic curve $E/\mathbb{Q}$.
In [2], the author claims that the fields $\mathbb{Q}(D_4^\infty)$ defined in the paper and the compositum of all $D_4$ extensions of $\mathbb{Q}$ coincide. The proof of this claim depends on a misreading of a celebrated result by Shafarevich. The purpose is to salvage the main results of [2]. That is, the classification of torsion structures of $E$ defined over $\mathbb{Q}$ when base changed to the compositum of all $D_4$ extensions of $\mathbb{Q}$ main results of [2]. All the main results in [2] are still correct except that we are no longer able to prove that these two fields are equal.
For a positive integer $N$ and $\mathbb{A}$ a subset of $\mathbb{Q}$, let $\mathbb{A}$-$\mathcal{KS}(N)$ denote the set of $α=\dfrac{α_{1}}{α_{2}}\in \mathbb{A}\setminus \{0,N\}$ verifying $α_{2}r-α_{1}$ divides $α_{2}N-α_{1}$ for every prime divisor $r$ of $N$. The set $\mathbb{A}$-$\mathcal{KS}(N)$ is called the set of $N$-Korselt bases in $\mathbb{A}$. Let $p, q$ be two distinct prime numbers. In this paper, we prove that each $pq$-Korselt base in $\mathbb{Z}\setminus\{ q+p-1\}$ generates other(s) in $\mathbb{Q}$-$\mathcal{KS}(pq)$. More precisely, we will prove that if $(\mathbb{Q}\setminu…
Permutation Rational Functions over Quadratic Extensions of Finite Fields
本文研究有限域二次扩张上的置换有理函数，利用与二次型相关的特征和确定特殊 $q$-二次多项式的零点数，并据此判定给定有理函数是否为置换。
The finite subgroups of ${\rm PGL}_2(\mathbb{C})$ are shown to be the only finite groups $G$ with this property: for some integer $r_0$ (depending on $G$), all Galois covers $X\rightarrow \mathbb{P}^1_{\mathbb{C}}$ of group $G$ can be obtained by pulling back those with at most $r_0$ branch points along non-constant rational maps $\mathbb{P}^1_{\mathbb{C}} \rightarrow \mathbb{P}^1_{\mathbb{C}}$. For $G\subset {\rm PGL}_2(\mathbb{C})$, it is in fact enough to pull back one well-chosen cover with at most $3$ branch points. A consequence of the converse for inverse Galois theory is that, for $G\n…
For a variety over certain topological rings $R$, like $\mathbb{Z}_p$ or $\mathbb{C}$, there is a well-studied way to topologize the $R$-points on the variety. In this paper, we generalize this definition to algebraic stacks. For an algebraic stack $\mathfrak{X}$ over many topological rings $R$, we define a topology on the isomorphism classes of $R$-points of $\mathfrak{X}$. We prove expected properties of the resulting topological spaces including functoriality. Then, we extend the definition to the case when $R$ is the ring of adeles of some global field. Finally, we use this last definition…
It is shown that $c=-29/16$ is the unique rational number of smallest denominator, and the unique rational number of smallest numerator, for which the map $f_c(x) = x^2+c$ has a rational periodic point of period $3$. Several arithmetic conditions on the set of all such rational numbers $c$ and the rational orbits of $f_c(x)$ are proved. A graph on the numerators of the rational $3$-periodic points of maps $f_c$ is considered which reflects connections between solutions of norm equations from the cubic field of discriminant $-23$.
Suppose that $E/\mathbb{Q}$ is an elliptic curve with a rational point $T$ of order $2$ and $α\in E(\mathbb{Q})$ is a point of infinite order. We consider the problem of determining the density of primes $p$ for which $α\in E(\mathbb{F}_{p})$ has odd order. This density is determined by the image of the arboreal Galois representation $τ_{E,2^{k}} : {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to {\rm AGL}_{2}(\mathbb{Z}/2^{k}\mathbb{Z})$. Assuming that $α$ is primitive (that is, neither $α$ nor $α+ T$ is twice a point over $\mathbb{Q}$) and that the image of the ordinary mod $2^{k}$ Galois rep…
The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny
本文通过计算 $p$-adic 体积积分和分析模曲线，确定了随机 Weierstrass 方程在 $p$-adic 数域上定义椭圆曲线具有非平凡 3 阶挠点或 3 次同源的概率，并探讨了 $\ell>3$ 素数的情形。
In this paper, we will define analogues of multiple zeta values by replacing the differential forms defining multiple zeta values with some $\mathbb{Q}$-rational differential forms on the Fermat curve $F_2$ of degree 2 and discuss their arithmetic properties. We also investigate a motivic structure of the motivic periods corresponding to our periods. However, in order to study them, the current theory for motivic zeta elements is insufficient, and it leads us to study the base extension of the space of the motivic periods $\mathcal{H}_4$ of level 4 and its Galois invariant part.
For an extension $K/\mathbb{F}_q(T)$ of the rational function field over a finite field, we introduce the notion of virtually $K$-rational Drinfeld modules as a function field analogue of $\mathbb{Q}$-curves. Our goal in this article is to prove that all virtually $K$-rational Drinfeld modules of rank two with no complex multiplication are parametrized up to isogeny by $K$-rational points of a quotient curve of the Drinfeld modular curve $Y_0(\mathfrak{n})$ with some square-free level $\mathfrak{n}$. This is an analogue of Elkies' well-known result on $\mathbb{Q}$-curves.
Let $f\in \mathbb{Q}(x)$ be a non-constant rational function. We consider "Waring's Problem for $f(x)$," i.e., whether every element of $\bbq$ can be written as a bounded sum of elements of $\{f(a)\mid a\in \mathbb{Q}\}$. For rational functions of degree $2$, we give necessary and sufficient conditions. For higher degrees, we prove that every polynomial of odd degree and every odd Laurent polynomial satisfies Waring's Problem. We also consider the "Easier Waring's Problem": whether every element of $\mathbb{Q}$ can be represented as a bounded sum of elements of $\{\pm f(a)\mid a\in \mathbb{Q}\…
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 a positive integer, $\mathbb{A}$ be a subset of $\mathbb{Q}$ and $α=\dfrac{α_{1}}{α_{2}}\in \mathbb{A}\setminus \{0,N\}$. $N$ is called an $α$-Korselt number (equivalently $α$ is said an $N$-Korselt base) if $α_{2}p-α_{1}$ divides $α_{2}N-α_{1}$ for every prime divisor $p$ of $N$. By the Korselt set of $N$ over $\mathbb{A}$, we mean the set $\mathbb{A}$-$\mathcal{KS}(N)$ of all $α\in \mathbb{A}\setminus \{0,N\}$ such that $N$ is an $α$-Korselt number. In this paper we determine explicitly for a given prime number $q$ and an integer $l\in \mathbb{N}\setminus \{0,1\}$, the set $\mathb…
For a positive integer $N$ and $\mathbb{A}$ a subset of $\mathbb{Q}$, let $\mathbb{A}$-$\mathcal{KS}(N)$ denote the set of $α=\dfrac{α_{1}}{α_{2}}\in \mathbb{A}\setminus \{0,N\}$ verifying $α_{2}p-α_{1}$ divides $α_{2}N-α_{1}$ for every prime divisor $p$ of $N$. The set $\mathbb{A}$-$\mathcal{KS}(N)$ is called the set of Korselt bases of $N$ in $\mathbb{A}$ or simply the $\mathbb{A}$-Korselt set of $N$. In this paper, we prove that for each squarefree composite number $N\in\mathbb{N}\setminus\{0,1\}$ the $\mathbb{Q}$-Korselt set of $N$ is finite where we provide an upper and lower bounds for e…
We show that, if a sequence of non-zero polynomials in $\mathbb{Z}[X_1,X_2]$ take small values at translates of a fixed point $(ξ,η)$ by multiples of a fixed rational point within the group $\mathbb{C}\times\mathbb{C}^*$, then $ξ$ and $η$ are both algebraic over $\mathbb{Q}$. The precise statement involves growth conditions on the degree and norm of these polynomials as well as on their absolute values at these translates. It is essentially best possible in some range of the parameters.
Rational Preperiodic Points of Quadratic Rational Maps over $\mathbb{Q}$ with Nonabelian Automorphism Groups
本文研究二次有理映射在有理数域上非交换自同构群下的有理数周期点问题，证明了不存在周期N≥4的有理数周期点，并给出了周期1、2、3的参数化结果，同时证明了预周期点数量不超过6，最终分类出5种预周期点图谱。
Height moduli on cyclotomic stacks and counting elliptic curves over function fields
本文利用twisted stable maps和stacky height functions理论，构造了参数化cyclotomic stacks上固定高度有理点的有限型模空间，解释了函数域上椭圆曲线计数中的低阶主项，回答了Venkatesh的问题。
Let $μ_{q+1}$ denote the set of $(q+1)$-th roots of unity in $\mathbb{F}_{q^2 }$. We construct permutation polynomials over $\mathbb{F}_{q^2}$ by using rational functions of any degree that induce bijections either on $μ_{q+1}$ or between $μ_{q+1}$ and $\mathbb{F}_q \cup \{\infty\}$. In particular, we generalize results from Zieve.
Let $\mathscr{E}\rightarrow\mathbb{P}^1_\mathbb{Q}$ be a non-trivial rational elliptic surface over $\mathbb{Q}$ with base $\mathbb{P}^1_\mathbb{Q}$ (with a section). We conjecture that any non-trivial elliptic surface has a Zariski-dense set of $\mathbb{Q}$-rational points. In this paper we work on solving the conjecture in case $\mathscr{E}$ is rational by means of geometric and analytic methods. First, we show that for $\mathscr{E}$ rational, the set $\mathscr{E}(\mathbb{Q})$ is Zariski-dense when $\mathscr{E}$ is isotrivial with non-zero $j$-invariant and when $\mathscr{E}$ is non-isotrivi…
Let $P\in\mathbb{P}_1(\mathbb{Q})$ be a periodic point for a monic polynomial with coefficients in $\mathbb{Z}$. With elementary techniques one sees that the minimal periodicity of $P$ is at most $2$. Recently we proved a generalization of this fact to the set of all rational functions defined over ${\mathbb{Q}}$ with good reduction everywhere (i.e. at any finite place of $\mathbb{Q}$). The set of monic polynomials with coefficients in $\mathbb{Z}$ can be characterized, up to conjugation by elements in PGL$_2({\mathbb{Z}})$, as the set of all rational functions defined over $\mathbb{Q}$ with a…
Rational points on smooth surfaces in $\mathbb{P}^3$ over finite fields
改进了第二作者关于有限域上光滑曲面有理点数目的界，并计算了达到或接近该界的曲面族的精确有理点数。
The number $N_9(5)$, the maximal number of $\mathbb{F}_9$-rational points on curves over $\mathbb{F}_9$ of genus $5$ is unknown, but it is known that $32 \le N_9(5)\le 35$. In this paper, we enumerate hyperelliptic curves and trigonal curves over $\mathbb{F}_3$ which have many $\mathbb{F}_9$-rational points (and $\mathbb{F}_3$-rational points), especially the maximal number of $\mathbb{F}_9$-rational points of those curves is $30$. Kudo-Harashita studied the nonhyperelliptic and nontrigonal case,where they found a new example of curves (over $\mathbb{F}_3$) of genus five which attains $32$ and…
Torsion of elliptic curves with rational $j$-invariant over the maximal elementary abelian 2-extension of $\mathbb{Q}$
本文针对定义在有理数域所有二次扩张合成域上、j不变量为非0或1728的有理数的椭圆曲线，完成了对其可能挠子群结构的分类。
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 study the Selmer varieties of smooth projective curves of genus at least two defined over $\mathbb{Q}$ which geometrically dominate a curve with CM Jacobian. We extend a result of Coates and Kim to show that Kim's non-abelian Chabauty method applies to such a curve. By combining this with results of Bogomolov-Tschinkel and Poonen on unramified correspondences, we deduce that any cover of $\mathbf{P}^1$ with solvable Galois group, and in particular any superelliptic curve over $\mathbb{Q}$, has only finitely many rational points over $\mathbb{Q}$.
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。
The problem of finding all possible extensions of a given rational diophantine quadruple to a rational diophantine quintuple is equivalent to the determination of the set of rational points on a certain curve of genus 5 that can be written as an intersection of three diagonal quadrics in $\mathbb P^4$. We discuss how one can (try to) determine the set of rational points on such a curve. We apply our approach to the original question in several cases. In particular, we show that Fermat's diophantine quadruple (1,3,8,120) can be extended to a rational diophantine quintuple in only one way, namel…
We give bounds on the degree of generation and relations of section rings associated to arbitrary $\mathbb{Q}$-divisors on projective spaces of all dimensions and Hirzebruch surfaces. For section rings of effective $\mathbb{Q}$-divisors on projective spaces, we find the best possible bound on the degrees of generators and relations.
In this paper we study the properties of the \emph{Triangular tree}, a complete tree of rational pairs introduced in \cite{cas}, in analogy with the main properties of the Farey tree (or Stern-Brocot tree). To our knowledge the Triangular tree is the first generalisation of the Farey tree constructed using the mediant operation. In particular we introduce a two-dimensional representation for the pairs in the tree, a coding which describes how to reach a pair by motions on the tree, and its description in terms of $SL(3,\mathbb{Z})$ matrices. The tree and the properties we study are then used t…
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 $\mathbb{Q}$.
Density of rational points on diagonal bidegree $(1,2)$ hypersurface in $\mathbb{P}^{s-1} \times \mathbb{P}^{s-1}$
本文针对 $\mathbb{P}^{s-1} \times \mathbb{P}^{s-1}$ 中对角双次数 $(1,2)$ 超曲面，在 $s \geq 7$ 时建立了有界反典范高度有理点数的渐近公式，从而证实了 Manin 猜想。
A Note On Transcendental Analytic Functions With Rational Coefficients Mapping $\mathbb{Q}$ Into Itself
本文研究将有理数集映射到自身的超越整函数问题，证明了对于任意 $t>0$，不存在分母增长满足 $O(q^{t})$ 且将 $\mathbb{Q}$ 映射到 $\mathbb{Q}$ 的超越整函数 $f\in\mathbb{Q}[[z]]$。
Let $K$ denote either $\mathbb{R}$ or $\mathbb{C}$. In this article, we introduce two new equivariants associated to a rational map $f\in K(z)$. These objects naturally live on a real hyperbolic space, and carry information about the action of $f$ on $\mathbb{P}^1(K)$. When $K=\mathbb{C}$ we relate the asymptotic behavior of these equivariants to the conformal barycenter of the measure of maximal entropy. We also give a complete description of these objects for rational maps of degree $d=1$. The constructions in this article are based on work of Rumely in the context of rational maps over non-…
This article examines dynamical systems on a class of K3 surfaces in $\mathbb{P}^{2} \times \mathbb{P}^{2}$ with an infinite automorphism group. In particular, this article develops an algorithm to find $\mathbb{Q}$-rational periodic points using information modulo $p$ for various primes $p$. The algorithm is applied to exhibit K3 surfaces with $\mathbb{Q}$-rational periodic points of primitive period $1,...,16$. A portion of the algorithm is then used to determine the Riemann zeta function modulo 3 of a particular K3 surface and find a family of K3 surfaces with Picard number two.
In this paper, we consider rational functions $f$ with some minor restrictions over the finite field $\mathbb{F}_{q^n},$ where $q=p^k$ for some prime $p$ and positive integer $k$. We establish a sufficient condition for the existence of a pair $(α,f(α))$ of primitive normal elements in $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}.$ Moreover, for $q=2^k$ and rational functions $f$ with quadratic numerators and denominators, we explicitly find that there are at most $55$ finite fields $\mathbb{F}_{q^n}$ in which such a pair $(α,f(α))$ of primitive normal elements may not exist.
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