Dragos Ghioca, Thomas J. Tucker
We prove that the local height of a point on a Drinfeld module can be computed by averaging the logarithm of the distance to that point over the torsion points of the module. This gives rise to a Drinfeld module analog of a weak version of Siegel's integral points theorem over number fields and to an analog of a theorem of Schinzel's regarding the order of a point modulo certain primes.
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We prove an equidistribution result for torsion points of Drinfeld modules of generic characteristic. We also show that similar equidistribution statements provide proofs for the Manin-Mumford and the Bogomolov conjectures for Drinfeld modules.
In this paper we prove a sparse equidistribution theorem for Gross points over the rational function field $\mathbb{F}_q(t)$. We apply this result to study the reduction map from CM Drinfeld modules to supersingular Drinfeld modules. Our proofs rely crucially on a period formula due to M. Papikian and F.-T. Wei/J. Yu, and a Lindelöf-type bound for central values of Rankin-Selberg $L$-functions associated to twists of automorphic forms of Drinfeld-type by ideal class group characters.
Equidistribution of Hecke Orbits on the Picard group of definite Shimura curves
本文利用自守方法及Drinfeld型尖点自守形式的系数界，证明了函数域上定志村曲线皮卡群中赫克轨道的等分布，特别是秩2超奇异Drinfeld模的赫克轨道等分布。
Duke for Drinfeld
本文在 Drinfeld–Stuhler 模曲线背景下证明了 CM 点的 Duke 均衡分布定理的函数域类比，将 Duke 定理和 Zhang 的均衡分布推广到 Drinfeld 情形，并处理了任意水平结构和正特征。
We compute the integral $p$-adic étale cohomology of Drinfeld symmetric spaces of any dimension. This refines the computation of the rational $p$-adic étale cohomology from Colmez-Dospinescu-Nizioł. The main tools are: the computation of the integral de Rham cohomology from CDN and the integral $p$-adic comparison theorems of Bhatt-Morrow-Scholze and Česnavičius-Koshikawa which replace the quasi-integral comparison theorem of Tsuji used in CDN.
Equidistribution of CM points on Shimura Curves and ternary theta series
本文证明全实域上志村曲线特殊纤维上CM点Galois轨道约化的等分布（含分裂和分歧情形，分歧情形用Cerednik-Drinfeld一致化模解释），通过关联权3/2希尔伯特模形式及分析傅里叶系数实现，并推导出André-Oort猜想积分版本的志村曲线情形。
We prove an equidistribution of signs for the Fourier coefficients of Hilbert modular forms of half-integral weight. Our study focuses on certain subfamilies of coefficients that are accessible via the Shimura correspondence. This is a generalization of the result of Inam and Wiese to the setting of totally real number fields
Let K be a global field of characteristic not 2. Let Z be a symmetric variety defined over K and S a finite set of places of K. We obtain counting and equidistribution results for the S-integral points of Z. Our results are effective when K is a number field.
We prove an analog of Siegel's theorem for integral points in the context of Drinfeld modules. The result holds for finitely generated submodules of the additive group over a function field of transcendence dimension 1.
In this paper, we discuss questions related to the oscillatory behavior and the equidistribution of signs for certain subfamilies of Fourier coefficients of integral weight newforms with a non-trivial nebentypus as well as Fourier coefficients of eigenforms of half-integral weight reachable by the Shimura correspondence.
In this paper we provide local test vector for Waldspurger's period integral, when the level of the representation $π_v$ is sufficiently large compared to the level of the character $Ω_v$ over quadratic extension, while allowing joint ramifications. The test vectors we shall use are variants of classical newforms, and the size of the resulting local integral is asymptotically the inverse of convexity bound for $L(Π\otimesΩ,1/2)$. Such test vectors are used to recover Gross-Prasad type test vectors. We also get vanishing result for local integral when using other test vectors. This phenomenon i…
Let $M=Γ\backslash\mathrm{PSL}(2,\mathbb{R})$ be a compact manifold, and let $f\in C^\infty(M)$ be a function of zero average. We use spectral methods to get uniform (i.e. independent of spectral gap) bounds for twisted averages of $f$ along long horocycle orbit segments. We apply this to obtain an equidistribution result for sparse subsets of horocycles on $M$.
We study the simultaneous sign change of Fourier coefficients of a pair of distinct normalized newforms of integral weight supported on primes power indices, we also prove some equidistribution results. Finally, we consider an analogous question for Fourier coefficients of a pair of half-integral weight Hecke eigenforms.
We prove that in the backward orbit of a non-preperiodic point under the action of a Drinfeld module of generic characteristic there exist at most finitely many points S-integral with respect to another nonpreperiodic point. This provides the answer (in positive characteristic) to a question raised by Sookdeo. We also prove that for each nontorsion point z, there exist at most finitely many torsion points which are S-integral with respect to z. This proves a question raised by Tucker and the author, and it gives the analogue of Ih's conjecture for Drinfeld modules.
Let f be a cusp form of weight k+1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato-Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp^2)}_p where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it f…
Asymptotics of integral points, equivariant compactifications and equidistributions for homogeneous spaces
针对线性代数群在Q上的齐次簇整点渐近计数问题，借助幂零流测度刚性将其约化为等变双有理几何问题，证明了特定条件下齐次空间的强Hardy-Littlewood性质及整点与体积渐近性的关系，并给出具体例子说明方法。
Let $E$ be an elliptic curve over an algebraically closed, complete, non-archimedean field $K$, and let ${\mathsf E}$ denote the Berkovich analytic space associated to $E/K$. We study the $μ$-equidistribution of finite subsets of $E(K)$, where $μ$ is a certain canonical unit Borel measure on ${\mathsf E}$. Our main result is an inequality bounding the error term when testing against a certain class of continuous functions on ${\mathsf E}$. We then give two applications to elliptic curves over global function fields: we prove a function field analogue of the Szpiro-Ullmo-Zhang equidistribution …
Let $f=\sum_{n=1}^{\infty}a(n)q^{n}\in S_{k+1/2}(N,χ_{0})$ be a non-zero cuspidal Hecke eigenform of weight $k+\frac{1}{2}$ and the trivial nebentypus $χ_{0}$ where the Fourier coefficients $a(n)$ are real. Bruinier and Kohnen conjectured that the signs of $a(n)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies $\{a(t n^{2})\}_{n}$ where $t$ is a squarefree integer such that $a(t)\neq 0$. Let $q$ and $d$ be natural numbers such that $(d,q)=1$. In this work, we show that $\{a(t n^{2})\}_{n}$ is equidistributed over any arithmetic p…
In this note, we improve earlier results towards the Bruinier-Kohnen sign equidistribution conjecture for half-integral weight modular eigenforms in terms of natural density by using a consequence of Halász' Theorem. Moreover, applying a result of Serre we remove all unproved assumptions.
Let $F$ be the field of rational functions on a smooth projective curve over a finite field, and let $π$ be an unramified cuspidal automorphic representation for $\mathrm{PGL}_2$ over $F$. We prove a variant of the formula of Yun and Zhang relating derivatives of the $L$-function of $π$ to the self-intersections of Heegner-Drinfeld cycles on moduli spaces of shtukas. In our variant, instead of a self-intersection, we compute the intersection pairing of Heegner-Drinfeld cycles coming from two different quadratic extensions of $F$, and relate the intersection to the $r$-th derivative of a produc…
We show that the module of integral points on a Drinfeld module satisfies a an analogue of Dirichlet's unit theorem, despite its failure to be finitely generated. As a consequence, we obtain a construction of a canonical finitely generated sub-module of the module of integral points. We use the results to give a precise formulation of a conjectural analogue of the class number formula.
Quantitative Equidistribution on Hyperbolic Surfaces and Arithmetic Applications
本文研究了双曲面上Wasserstein距离的定量等分布问题，并应用数论中的方法证明了Duke定理和Watson定理的上界。
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。
We give a simpler proof of an earlier result giving an asymptotic estimate for the number of integral matrices, in large balls, with a given monic integral irreducible polynomial as their common characteristic polynomial. The proof uses equidistributions of polynomial trajectories on SL(n,R)/SL(n,Z), which is a generalization of Ratner's theorem on equidistributions of unipotent trajectories. We also compute the exact constants appearing in the above mentioned asymptotic estimate.
We study the dynamics of stochastic families of rational maps on the projective line. As such families can be infinite and may not typically be defined over a single number field, we introduce the concept of generalized adelic measures, generalizing previous notions introduced by Favre and Rivera-Letelier and Mavraki and Ye. Generalized adelic measures are defined over the measure space of places of an algebraic closure of the rationals, using a framework established by Allcock and Vaaler. This turns our heights from sums over places into integrals. We prove an equidistribution result for gene…
We establish effective counting and equidistribution results for lattice points in families of domains in hyperbolic spaces, of any dimension and over any field. The domains we focus on are defined as product sets with respect to the Iwasawa decomposition. Several classical Diophantine problems can be reduced to counting lattice points in such domains, including distribution of shortest solution to the gcd equation, and angular distribution of primitive vectors in the plane. We give an explicit and effective solution to these problems, and extend them to imaginary quadratic number fields. Furt…
Integral torsion points on abelian varieties over function fields
本文在全局函数域上证明了关于阿贝尔变种上 torsion 点非 Zariski 密度的猜想，并建立了相关的 Tate-Voloch 类型定理，从而得到 Galois 轨道 torsion 点的对数等分布结果。
Mellin transform formulas for Drinfeld modules
本研究借助新框架推导Drinfeld模的对数公式，将其表为t-动机刚性解析平凡化乘积的动机映射取值，特殊化得到Goss L-函数特殊值表达式，并推广至带Tate扭的Drinfeld模，为特征p域提供了Hasse-Weil型ζ函数积分表示的类似物。
In analogy with the periods of abelian integrals of differentials of third kind for an elliptic curve defined over a number field, we introduce a notion of periods of third kind for a rank 2 Drinfeld Fq[t]-module rho defined over an algebraic function field and derive explicit formulae for them. When rho has complex multiplication by a separable extension, we prove the algebraic independence of rho-logarithms of algebraic points that are linearly independent over the CM field of rho. Together with the main result in [CP08], we completely determine all the algebraic relations among the periods …
Effective equidistribution of semisimple adelic periods and representations of quadratic forms
本文证明了半单 adelic 闭轨道的有效等分布定理，并应用于整二次型表示，改进余维数假设并提供有效界。
Let f be a classical holomorphic newform of level q and even weight k. We show that the pushforward to the full level modular curve of the mass of f equidistributes as qk -> infinity. This generalizes known results in the case that q is squarefree. We obtain a power savings in the rate of equidistribution as q becomes sufficiently "powerful" (far away from being squarefree), and in particular in the "depth aspect" as q traverses the powers of a fixed prime. We compare the difficulty of such equidistribution problems to that of corresponding subconvexity problems by deriving explicit extensions…
We use a dictionary between lattice point counting inside dilated d-dimensional ellipsoids (Euclidean counting) and counting of lifts of a closed horosphere that intersect a ball of increasing radius, to obtain two types of results. Firstly, via an $L^2$-integral error estimate for Euclidean counting, we prove effective equidistribution results for a family of expanding horospheres in the locally symmetric space SO(d) \ SL(d,R) / SL(d,Z). Secondly, we derive from uniform error estimates in Euclidean counting, error terms for counting SL(d,Z)-orbit points in a certain increasing family of subse…
Let $K$ be an algebraic function field with constant field ${\mathbb F}_q$. Fix a place $\infty$ of $K$ of degree $δ$ and let $A$ be the ring of elements of $K$ that are integral outside $\infty$. We give an explicit description of the elliptic points for the action of the Drinfeld modular group $G=GL_2(A)$ on the Drinfeld's upper half-plane $Ω$ and on the Drinfeld modular curve $G\!\setminus\!Ω$. It is known that under the {\it building map} elliptic points are mapped onto vertices of the {\it Bruhat-Tits tree} of $G$. We show how such vertices can be determined by a simple condition on their…
Equivariant Vector Bundles on the Drinfeld Upper Half Space over a Local Field of Positive Characteristic
本文将Orlik关于p-adic域的齐次向量丛整体截面工作推广至正特征局部域，通过子表示过滤和改编的函子刻画子商，并确定了K^×的局部解析与连续特征。
Anderson generating function of rank-one Drinfeld Module over rational function fields
本文研究了一阶Drinfeld模在有理函数域上的安德森生成函数，通过构造生成函数并将其与Carlitz周期关联，解决了无限地方度数大于1的理论空白，提出新的方法论进展。
A Criterion for Equidistribution along the $Ω$ Function over Polynomial Sequences with Applications
研究关于整系数齐次多项式\(P\)，建立沿\(\Omega(|P(n_1,...,n_d)|)\)的等分布准则，证明了相关定理变体，并在附录中证明了关于数域的相关猜想。
In this work, we begin to uncover the architecture of the general family of zeta functions and multiple zeta values as they appear in the theory of integrable systems and conformal field theory. One of the key steps in this process is to recognize the roles that zeta functions play in various arenas using transform methods. Other logical connections are provided by the the appearance of the Drinfeld associator, Hopf algebras, and techniques of conformal field theory and braid groups. These recurring themes are subtly linked in a vast scheme of a logically woven tapestry. An immediate applicati…
This is the first of a series of two papers in which we present a solution to Manin's Real Multiplication program -- an approach to Hilbert's 12th problem for real quadratic extensions of $\mathbb{Q}$ -- in positive characteristic, using quantum analogs of the exponential function and the modular invariant. In this first paper, we treat the problem of Hilbert class field generation. If $k=\mathbb{F}_{q}(T)$ and $k_{\infty}$ is the analytic completion of $k$, we introduce the quantum modular invariant \[ j^{\rm qt}: k_{\infty}\multimap k_{\infty}\] as a multivalued, modular invariant function. …
We prove that all points of a toroidal compactification lying over 0-dimensional cusps are rationally equivalent in the integral Chow group for most classical modular varieties (Siegel, Hilbert, orthogonal, Hermitian, quaternionic). This gives a generalization, and even strengthening, of the Manin-Drinfeld theorem in higher dimension from the viewpoint of algebraic cycles. The same result no longer holds for Picard modular varieties, but for them we prove that the difference of any two "special" boundary points, which are dense in the boundary, is torsion.
Euler factors of equivariant $L$--functions of Drinfeld modules and beyond
该研究针对Drinfeld模等变L函数的Euler因子在0处的恒等式问题，发展通用技术证明了任意Drinfeld模下的相关恒等式，并将方法推广至高维阿贝尔t模，为等变Tamagawa数公式提供支撑。
The geometric Satake equivalence for integral motives
本文研究了整数动机的几何萨塔克等价性，通过引入新的几何结构和等价性，完善了之前对分裂简约群的等价性结果，并给出了德林德勒修改的对偶群和文林格单形的塔恩克构造。
On the third kind periods for abelian $t$-modules
本文引入 abelian $t$-modules 的第三类周期概念并建立求值，推导出 Drinfeld modules 的显式公式，并证明任意秩 Drinfeld modules 三类周期的代数无关性。
Let $A=\mathbb{F}_q[T]$ be the polynomial ring over $\mathbb{F}_q$, and $F$ be the field of fractions of $A$. Let $ϕ$ be a Drinfeld $A$-module of rank $r\geq 2$ over $F$. For all but finitely many primes $\mathfrak{p}\lhd A$, one can reduce $ϕ$ modulo $\mathfrak{p}$ to obtain a Drinfeld $A$-module $ϕ\otimes\mathbb{F}_\mathfrak{p}$ of rank $r$ over $\mathbb{F}_\mathfrak{p}=A/\mathfrak{p}$. The endomorphism ring $\mathcal{E}_\mathfrak{p}=\mathrm{End}_{\mathbb{F}_\mathfrak{p}}(ϕ\otimes\mathbb{F}_\mathfrak{p})$ is an order in an imaginary field extension $K$ of $F$ of degree $r$. Let $\mathcal{O}_…
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…
Galois orbits of torsion points near atoral sets
本文将代数环面上挠点的伽罗瓦均匀分布推广到奇异检验函数，得到带幂节省的定量衰减估计，并据此证明了一类非环面除子的 Ih 整性有限性猜想。
Deformation theory for prismatic $G$-displays
本文发展了Bhatt-Scholze棱镜景上$G$-$μ$-display的形变理论并用棱镜$F$-规范加以解释，以此证明了具超特殊水平结构的整局部Shimura簇的局部可表示性与形式光滑性，并扩展了$p$-可除群的分类结果。
We compute the rational homotopy groups of the $K(n)$-local sphere for all heights $n$ and all primes $p$, verifying a prediction that goes back to the pioneering work of Morava in the early 1970s. More precisely, we show that the inclusion of the Witt vectors into the Lubin-Tate ring induces a split injection on continuous stabilizer cohomology with torsion cokernel of bounded exponent, thereby proving Hopkins' chromatic splitting conjecture and the vanishing conjecture of Beaudry-Goerss-Henn rationally. The key ingredients are the equivalence between the Lubin-Tate tower and the Drinfeld tow…
On integral local Shimura varieties
本文证明了对于任何阿贝尔类型的(G, μ)，当p≠2或p=2且G为A或C类型时，局部Shimura簇的构造猜想成立，并将其与局部Shimura品种联系起来。
We discuss Linnik's work on the distribution of integral solutions to $x^2+y^2+z^2 =d$, as $d$ goes to infinity. We give an exposition of Linnik's ergodic method; indeed, by using large-deviation results for random walks on expander graphs, we establish a refinement of his equidistribution theorem. We discuss the connection of these ideas with modern developments (ergodic theory on homogeneous spaces, $L$-functions).
We compute the statistics of $SL_{d}(\mathbb{Z})$ matrices lying on level sets of an integral polynomial defined on $SL_{d}(\mathbb{R})$, a result that is a variant of the well known theorem proved by Linnik about the equidistribution of radially projected integral vectors from a large sphere into the unit sphere. Using the above result we generalize the work of Aka, Einsiedler and Shapira in various directions. For example, we compute the joint distribution of the residue classes modulo $q$ and the properly normalized orthogonal lattices of primitive integral vectors lying on the level set $-…
Let $T$ be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over $T$, one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler's constant. In the final section, we evaluate more general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral -- over some polytopes that are higher-dimensional analog…
We formulate and prove the analogue of the Ramanujan Conjectures for modular forms of half-integral weight subject to some ramification restriction in the setting of a polynomial ring over a finite field. This is applied to give an effective solution to the problem of representations of elements of the ring by ternary quadratic forms. Our proof develops the theory of half-integral weight forms and Siegel's theta functions in this context as well as the analogue of an explicit Waldspurger formula. As in the case over the rationals, the half-integral weight Ramanujan Conjecture is in this way co…
We calculate the mean and variance of sums of the Möbius function and the indicator function of the squarefrees, in both short intervals and arithmetic progressions, in the context of the ring of polynomials over a finite field of $q$ elements, in the limit $q\to \infty$. We do this by relating the sums in question to certain matrix integrals over the unitary group, using recent equidistribution results due to N. Katz, and then by evaluating these integrals. In many cases our results mirror what is either known or conjectured for the corresponding problems involving sums over the integers, whi…
We give a group theoretic definition of "local models" as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a $p$-adic local field that are expected to model the singularities of integral models of Shimura varieties with parahoric level structure. Our local models are certain mixed characteristic degenerations of Grassmannian varieties; they are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and the second-named author to mixed characteristics and to the case of general (tamely ramified) reductive groups. We study the sing…
The recursive calculation of Selberg integrals by Aomoto and Terasoma using the Knizhnik-Zamolodchikov equation and the Drinfeld associator makes use of an auxiliary point and facilitates the recursive evaluation of string amplitudes at genus zero: open-string N-point amplitudes can be obtained from those at N-1 points. We establish a similar formalism at genus one, which allows the recursive calculation of genus-one Selberg integrals using an extra marked point in a differential equation of Knizhnik-Zamolodchikov-Bernard type. Hereby genus-one Selberg integrals are related to genus-zero Selbe…
Koksma's equidistribution theorem from 1935 states that for Lebesgue almost every $α>1$, the fractional parts of the geometric progression $(α^{n})_{n\geq1}$ are equidistributed modulo one. In the present paper we sharpen this result by showing that for almost every $α>1$, the correlations of all finite orders and hence the normalized gaps of $(α^{n})_{n\geq1}$ mod 1 have a Poissonian limit distribution, thereby resolving a conjecture of the two first named authors. While an earlier approach used probabilistic methods in the form of martingale approximation, our reasoning in the present paper …
We prove an analogue of a theorem of Eskin-Margulis-Mozes: suppose we are given a finite set of places $S$ over $\mathbb{Q}$ containing the archimedean place and excluding the prime $2$, an irrational isotropic form ${\mathbf q}$ of rank $n\geq 4$ on $\mathbb{Q}_S$, a product of $p$-adic intervals $I_p$, and a product $Ω$ of star-shaped sets. We show that unless $n=4$ and ${\mathbf q}$ is split in at least one place, the number of $S$-integral vectors ${\mathbf v} \in {\mathsf{T}} Ω$ satisfying simultaneously ${\mathbf q}( {\mathbf v} ) \in I_p$ for $p \in S$ is asymptotically given by $$ λ({\…
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