WonTae Hwang, Keunyoung Jeong
We count the number of isomorphism classes of degree $d$-twists of some polarized abelian varieties over finite fields of odd prime dimension. This can be seen as a higher dimensional analogue of the counting problem for elliptic curves case.
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The goal of this note is to present a combinatorial mechanism for counting certain objects associated to a variety X defined over a finite field. The basic example is that of counting conjugacy classes in GL_n(F_q), where X is the multiplicative group.
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.
In [17], we proved a structure theorem on the Mordell-Weil group of abelian varieties over function fields that arise as the twists of abelian varieties by the cyclic covers of projective varieties in terms of the Prym varieties associated with covers. In this paper, we provide an explicit way to construct the abelian varieties with large ranks over the higher dimension function fields. To do so, we apply the above-mentioned theorem to the twists of Albanese varieties of the cyclic multiple planes.
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 $V$.
Point Counting on Igusa Varieties for function fields
本文研究了全局 $\Gscr$-shtuka 模栈上的 Igusa 簇，计算了其上同调上的 Hecke 作用，并证明了局部 $G$-shtuka 的新结果及其在 Barsotti-Tate 群和 Shimura 簇中的应用。
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.
Let $\cal A$ be a maximal (or more generally a hereditary) order in a central simple algebra over a global field $F$ of positive characteristic. We study the reduction of the modular scheme of $\cal A$-elliptic sheaves at all places of $F$. We show that some of these varieties - for different $\cal A$ - are twists of each other. This result can be viewed as a global analogue of the Cherednik-Drinfeld theorem for these varieties.
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。
The present note is devoted to an amendment to a recent paper of Ellenberg, Lawrence and Venkatesh. Roughly speaking, the main result here states the subpolynomial growth of the number of integral points with bounded height of a variety over a number field whose fundamental group is large. Such an improvement, i.e. requiring large fundamental group as opposed to the existence of a geometric variation of pure Hodge structures, was already asked in op.cit..
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。
We present a deterministic polynomial time algorithm for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic. One consequence of this result is an efficient method for computing the order of the group of rational points on the Jacobian of a smooth geometrically connected projective curve over a finite field of small characteristic.
Igusa varieties are algebraic varieties that arise in the study of special fibers of Shimura varieties, and have demonstrated many applications in the Langlands program via a Langlands-Kottwitz style point-counting formula due to Shin in the case of PEL type. In this paper we formulate and prove an analogue of the Langlands-Rapoport conjecture for Igusa varieties of Hodge type, building off the work of Kisin in the case of Shimura varieties. We then use this description of the points to derive a point-counting formula for Igusa varieties of Hodge type, generalizing the fomula in PEL type of Sh…
Torsion and twists of abelian varieties
该研究探讨具好约化阿贝尔簇扭形的挠子群，证明相关二次扩张分歧素数定理，并指出满足Heegner假设的特定虚二次域上有理椭圆曲线无奇阶挠率增长。
Let $A$ and $A'$ be abelian varieties defined over a number field $k$ of dimension $g\geq 1$. For $g\leq 3$, we show that the following local-global principle holds: $A$ and $A'$ are quadratic twists of each other if and only if, for almost all primes $\mathfrak p$ of $k$ of good reduction for $A$ and $A'$, the reductions $A_{\mathfrak p}$ and $A'_{\mathfrak p}$ are quadratic twists of each other. This result is known when $g=1$, in which case it has appeared in works by Kings, Rajan, Ramakrishnan, and Serre. We provide an example that violates this local-global principle in dimension $g=4$.
A counting argument for the geometric Bombieri-Lang conjecture on ramified covers of abelian varieties
本文通过新的计数论证，证明了有理函数域上阿贝尔簇的分支覆盖满足几何Bombieri-Lang猜想，扩展了Xie-Yuan的结果。
We give some easy necessary and sufficient criteria for twists of abelian varieties by Artin representations to be simple.
Let $S$ be a Shimura variety and let $h$ be a Weil height function on $S$. We conjecture that the heights of special points in $S$ are discriminant negligible. Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In the case of Shimura varieties of abelian type, the height bound holds by the recently proved averaged Colmez formula, and our theorem gives a new proof of Tsimerman's Galois lower bound in this case. The main novelty is that our approach avoids the u…
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。
Let r > 0 be an integer. We present a sufficient condition for an abelian variety A over a number field k to have infinitely many quadratic twists of rank at least r, in terms of density properties of rational points on the Kummer variety Km(A^r) of the r-fold product of A with itself. One consequence of our results is the following. Fix an abelian variety A over k, and suppose that for some r > 0 the Brauer-Manin obstruction to weak approximation on the Kummer variety Km(A^r) is the only one. Then A has a quadratic twist of rank at least r. Hence if the Brauer-Manin obstruction is the only on…
Selmer ranks in twists of CM abelian varieties
本研究证明CM阿贝尔簇的若干p次扭变族中Selmer秩服从辛或酉分布，借此推出含本原p次单位根的数域上扭变费马曲线大多无解，并探讨了素数次循环扩张中的秩增长。
We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and $q$-analogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo the action of the $σ$-centralizer group to the Mirkovic-Vilonen basis of a certain weight space of a representation of the Langlands dual group.
Counting Polynomial-type Exceptional Units on Algebraic Varieties over Number Fields
本文研究了代数数域上代数簇上的多项式型例外单位的计数问题，通过中国剩余定理和Hensel提升技术，推导出在良好降解条件下精确计数公式，并利用Lang-Weil不等式建立渐进行为估计，特别是对二次度数以下的簇改进了误差项。
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。
If $V$ is a commutative algebraic group over a field $k$, $O$ is a commutative ring that acts on $V$, and $I$ is a finitely generated free $O$-module with a right action of the absolute Galois group of $k$, then there is a commutative algebraic group $I \otimes_O V$ over $k$, which is a twist of a power of $V$. These group varieties have applications to cryptography (in the cases of abelian varieties and algebraic tori over finite fields) and to the arithmetic of abelian varieties over number fields. For purposes of such applications we devote this article to making explicit this tensor produc…
This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number $\sqrt{q}$. This establishes a key step that one may extend our previous explicit calculations of superspecial abelian surfaces to those of supersingular abelian surfaces.The second part is to introduce the notion of ge…
We prove that for any two integers $g\geq 4$ and $g'\leq 2g-1$, there exist abelian varieties over $\overline{\mathbb{Q}}$ which are not quotients of a Jacobian of dimension $g'$. Our method in fact proves that most Abelian varieties satisfy this property, when counting by height relative to a fixed finite map to projective space.
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。
Ranks of abelian varieties in cyclotomic twist families
该研究在伽罗瓦作用可约假设下，给出分圆扭族中阿贝尔簇Mordell--Weil群平均秩的界，结合一致Mordell--Lang猜想得到相关曲线扭族有理点个数的近一致界，核心技术是确定3-同源Selmer群的平均大小。
Counting matrix points on certain varieties over finite fields
本文将有限域超几何函数方法推广到矩阵点计数，构造矩阵椭圆曲线及 K3 曲面族的计数公式，并证明误差项的 Sato-Tate 分布。
We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we determine the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun--Mazur--Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich--Tate group (if finite) may have order twice a square. In particular, the statistics for parities of 2-Selmer ranks and 2-infinity Selmer ranks need no longer agree…
On the local-global principle for twists of abelian varieties and Galois representations
本文研究了阿贝尔簇扭的局部-全局原理，定义并研究了相关的上同调集，并证明了其有限性和平凡条件。
Let $F$ be a totally real number field and $A/F$ a principally polarized abelian variety with real multiplication by the ring of integers $\mathcal{O}$ of a totally real field. Assuming $A$ admits an $\mathcal{O}$-linear 3-isogeny over $F$, we prove that a positive proportion of the quadratic twists $A_d$ have rank 0. We also prove that a positive proportion of $A_d$ have rank $\dim A$, assuming the Tate-Shafarevich groups are finite. If $A$ is the Jacobian of a hyperelliptic curve $C$, we deduce that a positive proportion of twists $C_d$ have no rational points other than those fixed by the h…
Power Savings for Counting (Twisted) Abelian Extensions of Number Fields
本文研究了数域阿贝尔扩张的计数问题，证明了误差项具有显著的幂次节省，并在假设广义Lindelöf假设下达到平方根幂次节省，同时揭示了部分未知的低阶项结构。
In this paper, we study the distribution of integral points on parametric families of affine homogeneous varieties. By the work of Borel and Harish-Chandra, the set of integral points on each such variety consists of finitely many orbits of arithmetic groups, and we establish an asymptotic formula (on average) for the number of the orbits indexed by their Siegel weights. In particular, we deduce asymptotic formulas for the number of inequivalent integral representations by decomposable forms and by norm forms in division algebras, and for the weighted number of equivalence classes of integral …
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。
GCD and ABC inequalities for abelian varieties over function fields
该研究证明了特征0函数域上阿贝尔簇有理点的GCD不等式与几何abc猜想的特殊情形，采用三步证明模式，借助Xie-Yuan高度等工具解决了Buium的相关猜想。
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 …
Counting abelian varieties over finite fields via Frobenius densities
本文通过为每个有理数位引入Frobenius密度因子，给出有限域上主极化阿贝尔簇同源类大小的质量公式，其推导基于Kottwitz公式和辛相似群上的测度分析，避免了对类数的直接计算。
Let G be a semisimple group over rational numbers and H is a subgroup over rational numbers. Given a representation of G and an integral vector x whose stabilizer is equal to H. In this paper we investigate the asymptotic of integral points on Gx with bounded height. We find its asymptotic up to an implicit constant when H is large in G but we allow the presence of intermediate subgroups. This is achieved by a novel combination of two equidistribution results in two different settings: one is that of Eskin, Mozes and Shah on a Lie group modulo a lattice and the other one is a result of Chamber…
In a previous paper {GN2} an effective solution of the lattice point counting problem in general domains in semisimple S-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action of G on G/Gamma, and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in {NS} and use them to establish several useful consequences of this property, including : Effective upper bounds on lifting for solutions of congruences in affine homogeneous v…
Let $A$ be an abelian variety over a number field $F$ and let $p$ be a prime. Cohen-Lenstra-Delaunay-style heuristics predict that the Tate-Shafarevich group of $A_s$ should contain an element of order $p$ for a positive proportion of quadratic twists $A_s$ of $A$. We give a general method to prove instances of this conjecture by exploiting independent isogenies of $A$. For each prime $p$, there is a large class of elliptic curves for which our method shows that a positive proportion of quadratic twists have nontrivial $p$-torsion in their Tate-Shafarevich groups. In particular, when the modul…
For an abelian variety $A$ over a number field $F$, we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption that the multiplication-by-3 isogeny on $A$ factors as a composition of 3-isogenies over $F$. This is the first such boundedness result for an absolutely simple abelian variety $A$ of dimension greater than one. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension one, we deduce that if $E/F$ is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded.…
We show that there exists a connection between two types of objects: some kind of resultantal varieties over C, from one side, and varieties of twists of the tensor powers of the Carlitz module such that the order of 0 of its L-functions at infinity is a constant, from another side. Obtained results are only a starting point of a general theory. We can expect that it will be possible to prove that the order of 0 of these L-functions at 1 (i.e. the analytic rank of a twist) is not bounded --- this is the function field case analog of the famous conjecture on non-boundedness of rank of twists of…
We extend recent orbit counts for finitely generated semigroups acting on $\mathbb{P}^N$ to certain infinitely generated, polarized semigroups acting on projective varieties. We then apply these results to semigroup orbits generated by some infinite sets of unicritical polynomials.
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.
Consider an irreducible, admissible representation $π$ of GL(2,$F$) whose restriction to GL(2,$F)^+$ breaks up as a sum of two irreducible representations $π_+ + π_-$. If $π=r_θ$, the Weil representation of GL(2,$F$) attached to a character $θ$ of $K^*$ which does not factor through the norm map from $K$ to $F$, then $χ\in \hat{K^*}$ with $(χ>. θ^{-1})|_{F^{*}}=ω_{K/F}$ occurs in ${r_θ}_+$ if and only if $ε(θχ^{-1},ψ_0)=ε(\bar θχ^{-1},ψ_0)=1$ and in ${r_θ}_-$ if and only if both the epsilon factors are -1. But given a conductor $n$, can we say precisely how many such $χ$ will appear in $π$? We…
In this short note we present an elementary proof of the Ax-Lindemann-Weierstrass theorem for abelian and semi-abelian varieties. The proof uses ideas of Pila, Ullmo, Yafaev, Zannier and is based on basic properties of sets definable in o-minimal structures. It does not use the Pila-Wilkie counting theorem.
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