The Sato-Tate Conjecture on Average for Small Angles

Sato-Tate Distributions on Abelian Surfaces

We prove a few new cases of the Sato-Tate conjecture for abelian surfaces, using a new automorphy theorem of Allen et al. Then in the unproven cases, we use partial results to describe nontrivial asymptotics on the trace of Frobenius, and prove their optimality given current knowledge.

Sato-Tate equidistribution of certain families of Artin L-functions

We study various families of Artin $L$-functions attached to geometric parametrizations of number fields. In each case we find the Sato-Tate measure of the family and determine the symmetry type of the distribution of the low-lying zeros.

Equidistribution, L-functions, and Sato-Tate groups

In this expository note, we present an approach to the generalization of Serre of the Sato-Tate Conjecture. Most of its content is taken from Serre's original references. However, we provide a few new examples and supply references to recent progress developed in the area.

Sato-Tate Distributions of $y^2=x^p-1$ and $y^2=x^{2p}-1$

We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form $$ y^2=x^p-1 \text{ and } y^2=x^{2p}-1,$$ where $p$ is an odd prime. Our results rely on the fact the Jacobians of these curves are nondegenerate, a fact that we prove in the paper. Furthermore, we compute moment statistics associated to the Sato-Tate groups. These moment statistics can be used to verify the equidistribution statement of the generalized Sato-Tate conjecture by comparing them to moment statistics obtained for the traces in the normalized $L$-polynomials of th…

Effective forms of the Sato--Tate conjecture

We prove effective forms of the Sato-Tate conjecture for holomorphic cuspidal newforms which improve on the author's previous work (solo and joint with Lemke Oliver). We also prove an effective form of the joint Sato-Tate distribution for two twist-inequivalent newforms. Our results are unconditional because of recent work of Newton and Thorne.

关于由椭圆曲线乘积产生的曲面的有效Sato-Tate分布

On Effective Sato-Tate Distributions for Surfaces Arising from Products of Elliptic Curves

本文证明了两族由椭圆曲线乘积产生的曲面（单参数K3曲面和双二次曲面）的Sato-Tate分布，并给出无条件有效误差界，同时完善了Thorner关于联合Sato-Tate分布的工作。

Pseudorandomness of Sato-Tate Distributions for Elliptic Curves

In this paper we propose conjectures that assert that, the sequence of Frobenius angles of a given elliptic curve over $\mathbf{Q}$ without complex multiplication is pseudorandom, in other words that the Frobenius angles are statistically independently distributed with respect to the Sato-Tate measure. Numerical evidences are presented to support the conjectures.

A remark on the component group of the Sato-Tate group

In this paper, we give a complete characterization of the component group of the Sato-Tate group of an abelian variety $A$ of arbitrary dimension, defined over a number field $K,$ in terms of the connectedness of the Lefschetz group associated to $A.$

On the Sato-Tate conjecture for non-generic abelian surfaces

We prove many non-generic cases of the Sato-Tate conjecture for abelian surfaces as formulated by Fite, Kedlaya, Rotger and Sutherland, using the potential automorphy theorems of Barnet-Lamb, Gee, Geraghty and Taylor.

The Sato-Tate conjecture for modular forms of weight 3

We prove a natural analogue of the Sato-Tate conjecture for modular forms of weight 2 or 3 whose associated automorphic representations are a twist of the Steinberg representation at some finite place.

The Sato-Tate law for Drinfeld modules

We prove an analogue of the Sato-Tate conjecture for Drinfeld modules. Using ideas of Drinfeld, J.-K. Yu showed that Drinfeld modules satisfy some Sato-Tate law, but did not describe the actual law. More precisely, for a Drinfeld module ϕdefined over a field L, he constructs a continuous representation ρ_\infty : W_L \to D^* of the Weil group of L into a certain division algebra, which encodes the Sato-Tate law. When the Drinfeld module has generic characteristic and L is finitely generated, we shall describe the image of this representation up to commensurability. As an application, we give i…

潜在为 $\ m{GL}_2$ 型的阿贝尔曲面的 Chebotarev-Sato-Tate 分布

Chebotarev-Sato-Tate distribution for abelian surfaces potentially of $\rm{GL}_2$-type

研究提出阿贝尔簇的混合 Chebotarev-Sato-Tate 猜想，并借助潜在自同构定理完成若干特殊情形的证明。

Arithmetic invariants from Sato--Tate moments

We give some arithmetic-geometric interpretations of the moments M_2[a_1], M_1[a_2], and M_1[s_2] of the Sato-Tate group of an abelian variety A defined over a number field by relating them to the ranks of the endomorphism ring and Néron-Severi group of A.

The Sato-Tate conjecture and Nagao's conjecture

Nagao's conjecture relates the rank of an elliptic surface to a limit formula arising from a weighted average of fibral Frobenius traces, and it is further generalized for smooth irreducible projective surfaces by M. Hindry and A. Pacheco. We show that the Sato-Tate conjecture based on the random matrix model implies Nagao's conjecture for certain twist families of elliptic curves and hyperelliptic curves.

Deformations of Galois Representations and the Theorems of Sato-Tate, Lang-Trotter and others

We construct infinitely ramified Galois representations $ρ$ such that the $a_l (ρ)$'s have distributions in contrast to the statements of Sato-Tate, Lang-Trotter and others. Using similar methods we deform a residual Galois representation for number fields and obtain an infinitely ramified representation with very large image, generalising a result of Ramakrishna.

Character theory approach to Sato-Tate groups

In this article, we propose to use the character theory of compact Lie groups and their orthogonality relations for the study of Frobenius distribution and Sato-Tate groups. The results show the advantages of this new approach in several aspects. With samples of Frobenius ranging in size much smaller than the moment statistic approach, we obtain very good approximation to the expected values of these orthogonality relations, which give useful information about the underlying Sato-Tate groups and strong evidence of the correctness of the generalized Sato-Tate conjecture. In fact, $2^{10}$ to $2…

The Sato-Tate Distribution in Thin Parametric Families of Elliptic Curves

We obtain new results concerning the Sato-Tate conjecture on the distribution of Frobenius traces over single and double parametric families of elliptic curves. We consider these curves for values of parameters having prescribed arithmetic structure: product sets, geometric progressions, and most significantly prime numbers. In particular, some families are much thinner than the ones previously studied.

Sato-Tate groups of y^2=x^8+c and y^2=x^7-cx

We consider the distribution of normalized Frobenius traces for two families of genus 3 hyperelliptic curves over Q that have large automorphism groups: y^2=x^8+c and y^2=x^7-cx with c in Q*. We give efficient algorithms to compute the trace of Frobenius for curves in these families at primes of good reduction. Using data generated by these algorithms, we obtain a heuristic description of the Sato-Tate groups that arise, both generically and for particular values of c. We then prove that these heuristic descriptions are correct by explicitly computing the Sato-Tate groups via the correspondenc…

The error term in the Sato-Tate theorem of Birch

We establish an error term in the Sato-Tate theorem of Birch. That is, for $p$ prime, $q=p^r$ we show that $\#\{ (a,b) \in \mathbb{F}_q^2 : θ_{a,b}\in I\} =μ_{ST}(I)q^2 + O_r(q^{7/4})$ for any interval $I\subseteq[0,π]$ where for an elliptic curve $E: y^2= x^3 +ax +b$, the quantity $θ_{a,b}$ is defined by $2\sqrt{q}\cosθ_{a,b} = q+1-E(\mathbb{F}_q)$ and $μ_{ST}(I)$ denotes the Sato-Tate measure of the interval $I$.

Expected Mordell-Weil rank heuristics through Sato-Tate, Birch and Swinnerton-Dyer conjectures

We present an heuristic argument for the prediction of expected Mordell-Weil rank of elliptic curves over number fields, using Birch and Swinnerton-Dyer's original conjecture and Sato-Tate conjectures. We do calculations in some cases and raise questions about their relations, if any, with the predictions of various average rank models that have been considered.

Sato-Tate Distributions of Catalan Curves

For distinct odd primes $p$ and $q$, we define the Catalan curve $C_{p,q}$ by the affine equation $y^q=x^p-1$. In this article we construct the Sato-Tate groups of the Jacobians in order to study the limiting distributions of coefficients of their normalized L-polynomials.Catalan Jacobians are nondegenerate and simple with noncyclic Galois groups (of the endomorphism fields over $\mathbb Q$), thus making them interesting varieties to study in the context of Sato-Tate groups. We compute both statistical and numerical moments for the limiting distributions. Lastly, we determine the Galois endomo…

Weighted Sato-Tate Vertical Distribution of the Satake Parameter of Maass Forms on PGL(N)

We formulate a conjectured orthogonality relation between the Fourier coefficients of Maass forms on PGL(N) for N>=2. Based on the work of Goldfeld-Kontorovich and Blomer for N=3, and on our conjecture for N>=4, we prove a weighted vertical equidistribution theorem (with respect to the generalized Sato-Tate measure) for the Satake parameter of Maass forms at a finite prime. For N=3, the rate of convergence for the equidistribution theorem is obtained.

Sato--Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height

We obtain asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of the elliptic curve $$ \E_{a,b} : Y^2 = X^3 + aX + b $$ satisfies certain ``natural'' properties, on average over integers $a$ and $b$ with $|a|\le A$ and $|b| \le B$, where $A$ and $B$ are small relative to $x$. Specifically, we investigate behavior with respect to the Sato--Tate conjecture, cyclicity, and divisibility of the number of points by a fixed integer $m$.

An algebraic Sato-Tate group and Sato-Tate conjecture

We make explicit a construction of Serre giving a definition of an algebraic Sato-Tate group associated to an abelian variety over a number field, which is conjecturally linked to the distribution of normalized L-factors as in the usual Sato-Tate conjecture for elliptic curves. The connected part of the algebraic Sato-Tate group is closely related to the Mumford-Tate group, but the group of components carries additional arithmetic information. We then check that in many cases where the Mumford-Tate group is completely determined by the endomorphisms of the abelian variety, the algebraic Sato-T…

Lang-Trotter and Sato-Tate Distributions in Single and Double Parametric Families of Elliptic Curves

We obtain new results concerning Lang-Trotter conjecture on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A.C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterized families of elliptic curves when the parameter runs through a set of rational numbers of bounded height. Some of the families we consider are much thinner than the ones previously studied.

Motivic Serre group, algebraic Sato-Tate group and Sato-Tate conjecture

We make explicit Serre's generalization of the Sato-Tate conjecture for motives, by expressing the construction in terms of fiber functors from the motivic category of absolute Hodge cycles into a suitable category of Hodge structures of odd weight. This extends the case of abelian varietes, which we treated in a previous paper. That description was used by Fite--Kedlaya--Rotger--Sutherland to classify Sato-Tate groups of abelian surfaces; the present description is used by Fite--Kedlaya--Sutherland to make a similar classification for certain motives of weight 3. We also give conditions under…

The Sato-Tate conjecture for Hilbert modular forms

We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of $\GL_2(\A_F)$, $F$ a totally real field, which are not of CM type. The argument is based on the potential automorphy techniques developed by Taylor et. al., but makes use of automorphy lifting theorems over ramified fields, together with a 'topological' argument with local deformation rings. In particular, we give a new proof of the conjecture for modular forms, which does not make use of poten…

GL(4) 的垂直 Sato-Tate 律

A vertical Sato-Tate law for GL(4)

本文建立了带显式误差项的 GL(4) 双加权垂直 Sato-Tate 律，通过将 Maass 尖点形式的正交关系推广至一般形式并去除了 Ramanujan 猜想假设。

On Sato--Tate distributions, extremal traces, and real multiplication in genus 2

The vertical Sato--Tate conjectures gives expected trace distributions for for families of curves. We develop exact expression for the distribution associated to degree-$4$ representations of $\mathrm{USp}(4)$, $\mathrm{SU}(2)\times\mathrm{SU}(2)$ and $\mathrm{SU}(2)$ in the neighborhood of the extremities of the Weil bound. As a consequence we derive qualitative distinctions between the extremal traces arising from generic genus-$2$ curves and genus-$2$ curves with real or quaternionic multiplication. In particular we show, in a specific sense, to what extent curves with real multiplication d…

Effective equidistribution and the Sato-Tate law for families of elliptic curves

Extending recent work of others, we provide effective bounds on the family of all elliptic curves and one-parameter families of elliptic curves modulo p (for p prime tending to infinity) obeying the Sato-Tate Law. We present two methods of proof. Both use the framework of Murty-Sinha; the first involves only knowledge of the moments of the Fourier coefficients of the L-functions and combinatorics, and saves a logarithm, while the second requires a Sato-Tate law. Our purpose is to illustrate how the caliber of the result depends on the error terms of the inputs and what combinatorics must be do…

Distribution of modular inverses and multiples of small integers and the Sato--Tate conjecture on average

We show that, for sufficiently large integers $m$ and $X$, for almost all $a =1, ..., m$ the ratios $a/x$ and the products $ax$, where $|x|\le X$, are very uniformly distributed in the residue ring modulo $m$. This extends some recent results of Garaev and Karatsuba. We apply this result to show that on average over $r$ and $s$, ranging over relatively short intervals, the distribution of Kloosterman sums $$ K_{r,s}(p) = \sum_{x=1}^{p-1} \exp(2 πi (rn + sn^{-1})/p), $$ for primes $p\le T$ is in accordance with the Sato--Tate conjecture.

Pair correlation statistics for Sato-Tate sequences

We investigate the pair correlation statistics for sequences arising from Hecke eigenvalues with respect to spaces of primitive modular cusp forms. We derive the average pair correlation function of Hecke angles lying in small subintervals of $[0,1]$. The averaging is done over non-CM newforms of weight $k$ with respect to $Γ_0(N).$ We also derive similar statistics for Hilbert modular forms and modular forms on hyperbolic 3-spaces.

Sato-Tate distributions of twists of the Fermat and the Klein quartics

We determine the limiting distribution of the normalized Euler factors of an abelian threefold A defined over a number field k when A is geometrically isogenous to the cube of a CM elliptic curve defined over k. As an application, we classify the Sato-Tate distributions of the Jacobians of twists of the Fermat and Klein quartics, obtaining 54 and 23, respectively, and 60 in total. We encounter a new phenomenon not visible in dimensions 1 or 2: the limiting distribution of the normalized Euler factors is not determined by the limiting distributions of their coefficients.

The Normal Distribution as a Limit of Generalized Sato-Tate Measures

With suitable order of limits, as p, m, and n all tend to infinity, the distribution of the normalized trace of Frobenius on H^1 of a "random" plane curve of degree n over the field with p^m elements, tends to a Gaussian distribution. The same is true of a "random" curve of genus g over the field with p^m elements.

Patterns of primes in the Sato-Tate conjecture

Fix a non-CM elliptic curve $E/\mathbb{Q}$, and let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius at $p$. The Sato-Tate conjecture gives the limiting distribution $μ_{ST}$ of $a_E(p)/(2\sqrt{p})$ within $[-1, 1]$. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval $I\subseteq [-1, 1]$, let $p_{I,n}$ denote the $n$th prime such that $a_E(p)/(2\sqrt{p})\in I$. We show $\liminf_{n\to\infty}(p_{I,n+m}-p_{I,n}) < \infty$ for all $m\ge 1$ for "most" intervals, and in particular, for all $I$ with $μ_{ST}(I)\ge 0.36$. Furthe…

Hecke 代数的有效生成与 Sato–Tate 型的显式估计

Effective generation of Hecke algebras and explicit estimates of Sato--Tate type

本文在相关 L 函数满足 Riemann 假设的前提下，给出显式有限 Hecke 算子集，用以生成高权、无平方因子水平模形式空间上的 Hecke 代数。

On the rank and the convergence rate towards the Sato-Tate measure

Let $A$ be an abelian variety defined over a number field and let $G$ denote its Sato-Tate group. Under the assumption of certain standard conjectures on $L$-functions attached to the irreducible representations of $G$, we study the convergence rate of any virtual selfdual character of $G$. We find that this convergence rate is dictated by several arithmetic invariants of $A$, such as its rank or its Sato-Tate group $G$. The results are consonant with some previous experimental observations, and we also provide additional numerical evidence consistent with them. The techniques that we use were…

广义Sato-Tate与二次剩余

Generalized Sato-Tate and quadratic residues

本文证明广义Sato-Tate猜想可用于精确刻画大素数模下连续二次剩余的分布规律。

Fields of definition of elliptic $k$-curves and the realizability of all genus 2 Sato--Tate groups over a number field

Let $A/\mathbb{Q}$ be an abelian variety of dimension $g\geq 1$ that is isogenous over $\overline{\mathbb{Q}}$ to $E^g$, where $E$ is an elliptic curve. If $E$ does not have complex multiplication (CM), by results of Ribet and Elkies concerning fields of definition of elliptic $\mathbb{Q}$-curves $E$ is isogenous to a curve defined over a polyquadratic extension of $\mathbb{Q}$. We show that one can adapt Ribet's methods to study the field of definition of $E$ up to isogeny also in the CM case. We find two applications of this analysis to the theory of Sato--Tate groups: First, we show that $1…

Maaß形式上GSp₄的加权垂直Sato-Tate定律

A weighted vertical Sato-Tate law for Maaß forms on $\rm{GSp}_4$

本文研究了GSp₄上自守形式的Satake参数加权Sato-Tate定律，结合Kuznetsov公式和局部积分计算，探讨了连续谱限制问题与双余类组合问题的关系。

算术级数中Frobenius迹的奇次矩与Sato–Tate猜想

Odd moments for the trace of Frobenius and the Sato--Tate conjecture in arithmetic progressions

该文研究算术级数限制下椭圆曲线Frobenius迹的矩，得到奇次矩与第零次矩之比的渐近行为，并据此证明其分布关于Sato–Tate测度等分布，为数论中相关猜想提供了新结果。

Fluctuations in the distribution of Hecke eigenvalues about the Sato-Tate measure

We study fluctuations in the distribution of families of $p$-th Fourier coefficients $a_f(p)$ of normalised holomorphic Hecke eigenforms $f$ of weight $k$ with respect to $SL_2(\mathbb{Z})$ as $k \to \infty$ and primes $p \to \infty.$ These families are known to be equidistributed with respect to the Sato-Tate measure. We consider a fixed interval $I \subset [-2,2]$ and derive the variance of the number of $a_f(p)$'s lying in $I$ as $p \to \infty$ and $k \to \infty$ (at a suitably fast rate). The number of $a_f(p)$'s lying in $I$ is shown to asymptotically follow a Gaussian distribution when a…

Towards the Sato-Tate Groups of Trinomial Hyperelliptic Curves

We consider the identity component of the Sato-Tate group of the Jacobian of curves of the form $$C_1\colon y^2=x^{2g+2}+c, C_2\colon y^2=x^{2g+1}+cx, C_3\colon y^2=x^{2g+1} +c,$$ where $g$ is the genus of the curve and $c\in\mathbb Q^*$ is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for $C_1$ curves of genus 4 and 5 and prove what the identity component of the Sato-Tate group is in each case. We then determine the splitting of Jacobians of higher genus $C_1$ curves by finding maps to lower genus curves and then…

On conjectures of Sato-Tate and Bruinier-Kohnen

This article covers three topics. (1) It establishes links between the density of certain subsets of the set of primes and related subsets of the set of natural numbers. (2) It extends previous results on a conjecture of Bruinier and Kohnen in three ways: the CM-case is included; under the assumption of the same error term as in previous work one obtains the result in terms of natural density instead of Dedekind-Dirichlet density; the latter type of density can already be achieved by an error term like in the prime number theorem. (3) It also provides a complete proof of Sato-Tate equidistribu…

Bianchi 模形式的 Ramanujan 与 Sato-Tate 猜想

The Ramanujan and Sato-Tate Conjectures for Bianchi modular forms

本文证明了 Bianchi 模形式及更一般的平行权正则代数尖点自守表示的 Ramanujan 与 Sato-Tate 猜想，其核心方法是通过一个新的潜在自同构定理处理二维相容 Galois 表示系的对称幂。

有效联合Sato-Tate分布与对称幂系数的符号变化

Effective Joint Sato-Tate Distribution and Sign Change of Symmetric Power Coefficients

本文研究了非CM新形式傅里叶系数的联合Sato-Tate分布及符号变化，扩展了Thorner的结果至更广泛的可测区域，提供了关于对称幂L函数系数的分布和符号行为的定量分析。

通过 Chebotarev--Sato--Tate 分布得到的多项式取值上算术函数的上界

Upper bounds for arithmetic functions over polynomial values via Chebotarev--Sato--Tate distributions

本文通过 Chebotarev--Sato--Tate 分布建立非负乘法函数在多元多项式取值处的对数上界，统一处理模形式等系数，并给出傅里叶系数的定量可除性结果。

Sato-Tate型测度中偏差的一个下界

A lower bound for the discrepancy in a Sato-Tate type measure

本文在适当条件下证明了偶整数权$k$、水平$N$的尖点形式空间$S_k(N)$的Petersson迹公式渐近式，据此给出水平不被8整除时Sato-Tate分布偏差的下界，推广了相关结果并应用于Omar和Mazhouda的类似分布。

Moments of the error term in the Sato-Tate law for elliptic curves

We derive new bounds for moments of the error in the Sato-Tate law over families of elliptic curves. Our estimates are stronger than those obtained by W.D. Banks and I.E. Shparlinski (arXiv:math/0609144) and L. Zhao and the fist-named author in (arXiv:math/0608318) for the first and second moments, but this comes at the cost of larger ranges of averaging. As applications, we deduce new almost-all results for the said errors and a conditional Central Limit Theorem on the distribution of these errors. Our method is different from those used in the above-mentioned papers and builds on recent work…

Motivic Serre group and Sato--Tate conjecture

This paper concerns the Algebraic Sato--Tate and Sato--Tate conjectures, based on Serre's original motivic formulation, with an eye towards explicit computations of Sato--Tate groups. We build on the algebraic framework for the Sato--Tate conjecture introduced in a previous paper, which used Deligne's motivic category for absolute Hodge cycles and was restricted to motives of odd weight. Here, we allow general weight and some other motivic categories, notably André's motivic category of motivated cycles; moreover, some results are also new in the odd weight case. The paper consists of two part…

垂直Sato-Tate的精细化

Refinements on vertical Sato-Tate

本研究通过刻画椭圆曲线Frobenius迹与计数分布的二维联合分布，并证明区间内曲线数量的强收敛性，对垂直Sato-Tate定理进行了精细化，其证明依赖于截断Gekeler乘积公式与有效Chebotarev密度定理。

Rankin-Selberg L-函数的例外零点与联合Sato-Tate分布

Exceptional zeros of Rankin-Selberg $L$-functions and joint Sato-Tate distributions

本文研究了Rankin-Selberg L-函数的例外零点问题，并建立了关于Hilbert cuspidal形式的Sato-Tate分布的最强大收敛率。

椭圆曲线高秩Zeta函数的Murmuration现象与Sato-Tate猜想

Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves

本研究将有理数域上椭圆曲线的Sato-Tate猜想与murmuration现象从阿贝尔Hasse-Weil Zeta函数推广至其p约化的非阿贝尔高秩Zeta函数，证明二者在该场景下同样成立。

Sato-Tate distributions of twists of y^2=x^5-x and y^2=x^6+1

We determine the limiting distribution of the normalized Euler factors of an abelian surface A defined over a number field k when A is isogenous to the square of an elliptic curve defined over k with complex multiplication. As an application, we prove the Sato-Tate Conjecture for Jacobians of Q-twists of the curves y^2=x^5-x and y^2=x^6+1, which give rise to 18 of the 34 possibilities for the Sato-Tate group of an abelian surface defined over Q. With twists of these two curves one encounters, in fact, all of the 18 possibilities for the Sato-Tate group of an abelian surface that is isogenous t…