Patrick Meisner
We study the moments of $\mbox{Tr}(Θ_χ)$ as $χ$ runs over Dirichlet characters defined over $\mathbb{F}_q[T]$ of fixed order $r$. In particular, we show that after an appropriate normalization, the $q$-limit of the power sum moments behave like the power sum moments of the group of unitary matrices multiplied by a weight function.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Recently E. Bombieri and N. M. Katz (2010) have demonstrated that several well-known results about the distribution of values of linear recurrence sequences lead to interesting statements for Frobenius traces of algebraic curves. Here we continue this line of study and establish the Möbius randomness law quantitatively for the normalised form of Frobenius traces.
Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In 1976, Lang and Trotter conjectured an asymptotic formula for the number $π_{E,r}(X)$ of primes $p \leq X$ of good reduction for which the Frobenius trace at $p$ associated to $E$ is equal to a given fixed integer $r$. We investigate elliptic curves $E$ over $\mathbb{Q}$ that have a missing Frobenius trace, i.e. for which the counting function $π_{E,r}(X)$ remains bounded as $X \rightarrow \infty$, for some $r \in \mathbb{Z}$. In particular, we classify all elliptic curves $E$ over $\mathbb{Q}(t)$ that have a missing Frobenius trace.
Poor man's transcendence for Frobenius traces of elliptic curves
本文证明了无复乘椭圆曲线的弗罗贝尼乌斯迹序列在有理数域上是超越的。
Hypergeometric functions over $\mathbb{F}_q$ and traces of Frobenius for elliptic curves
本文给出了某些椭圆曲线族的Frobenius自同态的迹与$\mathbb{F}_q$上的${_{2}}F_1$ -超几何函数的特殊值之间的显式关系，其中${_{2}}F_1$是$q \equiv 1 (\text{mod} 6)$和$q \equiv 1 (\text{mod} 4)$ .
Gaussian Hypergeometric Evaluations of Traces of Frobenius for Elliptic Curves
给出了满足$j(E)\neq 0,1728$和$q\equiv 1 \pmod{12}$的椭圆曲线$E$在$\mathbb{F}_q$上的Frobenius自同态的迹用高斯超几何级数的特殊值表示的公式.本文利用Fuselier的单参数曲线族的工作中介绍的方法，将$E$的Frobenius迹表示为曲线的$j$ -不变量和判别式的函数，而不是曲线的一个参数，这是曲线更内在的特征。
p-adic Gamma function and traces of Frobenius of elliptic curves
在\cite{mccarthy2}中，McCarthy利用有限域的Teichmüller特征标和$p$ -adic Γ函数的导数定义了一个函数$_{n}G_{n}[\cdots]$，并用$_{2}G_{2}[\cdots]$的特殊值表示了椭圆曲线的Frobenius迹.利用函数$_{2}G_{2}[\cdots]$建立了椭圆曲线的Frobenius迹的两种不同的表达式。结果得到了函数$_{2}G_{2}[\cdots]$在不同参数下的特殊值之间的两种关系。
We obtain distribution results for traces of Frobenius for various families of elliptic curves with respect to the Lang-Trotter conjecture, extremal primes, and the central limit theorem. This includes some generalisations and bounds related to the work of Sha-Shparlinski on the average Lang-Trotter conjecture for single-parametric families of elliptic curves and the work of various authors on the trace of Frobenius for primes in congruence classes. Some results are also obtained for modular forms.
Statistics for products of traces of high powers of the frobenius class of hyperelliptic curves
研究了固定有限域上亏格为g的超椭圆曲线的Frobenius类的高次幂迹的平均.我们证明了当亏格g增大时，这些乘积的极限期望等于曲线在酉辛群USp（2 g）上变化时的期望.我们还考虑了超椭圆曲线的Frobenius类特征相位的线性统计量的标度极限，并证明了它们的前几阶矩是高斯的。
The trace of Frobenius of elliptic curves and the $p$-adic gamma function
我们定义了一个函数的$p$ -adic的Γ函数，推广了作者早期的工作，在有限域上扩展超几何函数的$p$ -adic设置。证明了对于素数$p > 3$，$\mathbb{F}_p$上任何椭圆曲线的Frobenius迹，只要其$j$ -不变量不等于0或1728，都是该函数的一个特殊值.这推广了Fuselier和Lennon的结果，当$p \equiv 1 \pmod {12}$时，他们用$\mathbb{F}_p$上的超几何函数来评价Frobenius的迹。
Let $q$ be an odd prime power, $a \in \mathbb{F}_q[T]$ and $u \in \mathbb{F}_q^*$. Provided $q \geq 17$, we compute the average number of primes $p$ for which the characteristic polynomial of the Frobenius at $p$ is $X^2 - aX + up$ over a family of rank 2 Drinfeld $ \mathbb{F}_q[T]$-modules. Our results give asymptotic formulas in the $x$-limit.
Weighted averages of $p$-adic hypergeometric functions and traces of Frobenius of elliptic curves
研究一类一参数椭圆曲线的弗罗贝尼乌斯迹及其与p-进超几何函数的关系，通过加权平均特殊值和特殊值表达，推导出p-进超几何函数的求和恒等式和p-进欧拉-普夫夫变换。
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。
On the finite transcendence of Frobenius traces for abelian varieties over $\mathbb{Q}$
本文通过推广 Luca-Zudilin 的方法，证明了 $\mathbb{Q}$ 上椭圆曲线（无需复乘法假设）及若干主极化阿贝尔簇的 Frobenius 迹具有有限超越性。
Frobenius structure on hypergeometric equations, p-adic polygamma values and p-adic L-values
本文研究了超几何方程上的弗罗贝尼乌斯结构，通过p进polygamma函数推广了Kedlaya的公式，并将其与p进L值联系起来，应用于p进上同调的研究。
Frobenius Traces for Rank-2 Drinfeld Modules, Higher-Dimensional Galois Representations, and a Strong Multiplicity One Theorem in Positive Characteristic
本文证明了在有限多个地方外弗罗贝尼乌斯迹相等时，两类2阶非CM德林费尔德模的l-adic伽罗华表示同构。进一步，作者在弱维度假设下，展示在正特征局部域上，伽罗华表示在有限多个地方外弗罗贝尼乌斯迹相等强制同构仅在绝对不可约性假设下成立。
How Twist Class Redundancy Drives the Prediction of Traces of Frobenius of Elliptic Curves
本文探讨了二次扭类中的冗余如何影响椭圆曲线弗罗贝尼乌斯迹预测，提出通过独特扭类代表构建基准数据集以避免数据冗余影响。
Denote $Θ_C$ as the Frobenius class of a curve $C$ over the finite field $\mathbb{F}_q$. In this paper we determine the expected value of Tr$(Θ_C^n)$ where $C$ runs over all biquadratic curves when $q$ is fixed and $g$ tends to infinity. This extends work done by Rudnick and Chinis who separately looked at hyperelliptic curves and Bucur, Costa, David, Guerreiro and Lowry-Duda who looked at $\ell$-cyclic curves, for $\ell$ a prime, as well as cubic non-Galois curves.
We consider the expected value of $\mbox{Tr}(Θ_C^n)$ where $C$ runs over a thin family of $r$-cyclic covers of $\mathbb{P}^1_{\mathbb{F}_q}$ for any $r$. We obtain many lower order terms dependent on the divisors of $r$. We use these results to calculate the one-level density of the family and hypothesize a refined one-level density result.
Trace of Frobenius endomorphism of an elliptic curve with complex multiplication
设E是一条椭圆曲线，它与R作复乘，其中R是虚二次域K的判别式D<-4的阶.若素数p在与R相关的环类域上完全分解，则E在K的素理想P上有好约化，且存在正整数u和v使得4p=u^2-Dv^2.众所周知，E的模P约简的弗罗贝纽斯自同态的迹a_P的平方等于u^2。在R的类数为2或3，D被3，4或5整除的情况下，我们确定a_P=u还是a_P=-u。本文是作者预印本DMIS-RR-02- 5，2002的修订版。
Traces of high powers of the Frobenius class in the hyperelliptic ensemble
有限域上曲线的zeta函数可以用酉辛矩阵的特征多项式表示，称为曲线的Frobenius类。在大亏格极限下，计算了有限域上亏格为g的超椭圆曲线的集合的Frobenius类的n次幂迹的期望值，并将结果与酉辛群USp（2g）上相应的平均值进行了比较.我们能够计算幂n的平均值几乎高达4g，发现与随机矩阵结果一致，除了小n和n=2g。作为一个应用，我们计算的zeta函数的曲线，包括低阶项，检验函数的傅里叶变换支持（-2，2）的零点的一级密度。结果部分证实了Katz和Sarnak的一个猜想，即这个系综的低阶零点具有辛统计。
Traces of High Powers of the Frobenius Class in the Moduli Space of Hyperelliptic Curves
有限域上的曲线$C$的Zeta函数可以用酉矩阵$Θ_C$的特征多项式表示。在Rudnick工作的基础上，我们计算了有限域$\mathbb{F}_q$上亏格为$g$的超椭圆曲线的模空间上的$\mbox{tr}(Θ_C^n)$在大亏格极限下的期望值.作为应用，我们计算了当亏格趋于无穷大时，$C$在$\mathbb{F}_{q^n}$中的点数的期望值。我们还研究了小值$n$的两个预期值的偏倚。
On traces of Frobenius endomorphisms
我们利用V的所谓Serre C* 代数的不变量来计算有限域上投射簇V的点数。
Let $A$ be an abelian variety over $\mathbb{Q}$ of dimension $g$ such that the image of its associated absolute Galois representation $ρ_A$ is open in $\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. We investigate the arithmetic of the traces $a_{1, p}$ of the Frobenius at $p$ in $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ under $ρ_A$, modulo varying primes $p$. In particular, we obtain upper bounds for the counting function $\#\{p \leq x: a_{1, p} = t\}$ and we prove an Erdös-Kac type theorem for the number of prime factors of $a_{1, p}$. We also formulate a conjecture about the asymp…
Distributions of traces of Frobenius for smooth plane curves over finite fields
在以前的文章中，我们得到的非超椭圆亏格$3$曲线的Frobenius迹在小有限域上的分布数据。在目前的一个，我们给这些数据的启发式的解释，通过外推的平面曲线，其程度是小的相对于他们的有限基域的基数的轨迹分布的结果。特别是，我们的方法揭示了一些新的光分布的不对称性周围的平均值，这是有关的塞尔阻塞。
We explore an idea of Conrey and Li of expressing the Selberg trace formula as a Dirichlet series. We describe two applications, including an interpretation of the Selberg eigenvalue conjecture in terms of quadratic twists of certain Dirichlet series, and a formula for an arithmetically weighted sum of the complete symmetric square L-functions associated to cuspidal Maass newforms of squarefree level N>1.
Let $S$ be a smooth cubic surface over a finite field $\mathbb F_q$. It is known that $\#S(\mathbb F_q) = 1 + aq + q^2$ for some $a \in \{-2,-1,0,1,2,3,4,5,7\}$. Serre has asked which values of a can arise for a given $q$. Building on special cases treated by Swinnerton-Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton-Dyer's tables on cubic surfaces over finite fields.
We study the distribution of the Frobenius traces on $K3$ surfaces. We compare experimental data with the predictions made by the Sato--Tate conjecture, i.e. with the theoretical distributions derived from the theory of Lie groups assuming equidistribution. Our sample consists of generic $K3$ surfaces, as well as of such having real and complex multiplication. Each time, the theoretical density and the histogram obtained by counting points match in the range of visible accuracy. Thus, we report evidence for the Sato-Tate conjecture for the surfaces considered.
On the Frobenius fields of abelian varieties over number fields
研究数字段上非CM简单阿贝尔簇的弗罗贝尼乌斯域性质，证明满足特定条件的数域地方集的上狄利克雷密度为零，并在GRH假设下给出计数的幂次改进上界。
$p$-Adic hypergeometric functions and the trace of Frobenius of elliptic curves
本文研究$p$-进超几何函数，证明了${_4}G_4$与${_2}G_2$型函数的恒等式、高低阶函数的变换公式，并给出椭圆曲线Frobenius迹的超几何函数表示，推广了有限域相关结果的适用素数范围。
Character formula for Weil representations in terms of Frobenius traces
本文通过Frobenius迹研究Weil表示的特征公式，推广了局部域上阿贝尔簇上同调的确定性结果，并统一了p进域上的扩张条件。
Hypergeometric functions over F_p and relations to elliptic curves and modular forms
对于p与1 mod 12全等的素数，给出了一类j-不变量为1728/t的椭圆曲线上的Frobenius迹与F_p上一个特殊的2F_1-超几何函数的值之间的显式关系，并利用Frobenius迹确定了Hecke算子T_k（p）在权为k且水平为1的尖点形式空间上的迹公式.这导致拉马努金的τ函数的公式在超几何函数。
Let $g \geq 1$ be an integer and let $A/\mathbb{Q}$ be an abelian variety that is isogenous over $\mathbb{Q}$ to %the product $E_1 \times \ldots \times E_g$ of elliptic curves $E_1/\mathbb{Q}$, $\ldots$, $E_g/\mathbb{Q}$, without complex multiplication and pairwise non-isogenous over $\overline{\mathbb{Q}}$. a product of $g$ elliptic curves defined over $\mathbb{Q}$, pairwise non-isogenous over $\overline{\mathbb{Q}}$ and each without complex multiplication. %pairwise non-isogenous over $\overline{\mathbb{Q}}$. For an integer $t$ and a positive real number $x$, denote by $π_A(x, t)$ the number…
The stabilization of the Frobenius--Hecke traces on the intersection cohomology of orthogonal Shimura varieties
本文研究有理数域上特殊正交群的 Shimura 簇，证明并稳定化 Baily–Borel 紧化交上同调上 Frobenius–Hecke 迹的 Morel 公式，并借助自守表示内窥镜分类计算特殊情形的 Hasse–Weil ζ 函数。
Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves
研究有理数域上椭圆曲线约化最小韦尔斯特拉斯系数与弗罗贝尼乌斯迹的关系，用决策树模型发现可由特定素数处的迹及导体奇偶性恢复系数，还证明了新的显式公式，推断出系数由同构类决定。
Galois representations attached to moments of Kloosterman sums and conjectures of Evans
有限域的Kloosterman和是定义在$\Gm$上的某些局部系统的Frobenius迹函数。Kloosterman和的矩计算张量幂（或对称幂，外幂等）上同调的Frobenius迹。这些地方系统。我们表明，当$p$范围在所有素数，相应的Kloosterman和$\mathbb{F}_{p}$的时刻出现作为Frobenius迹上的连续$\ell$ -adic伽罗瓦表示，来自几何。我们还给出了这些伽罗瓦表示的分歧的界限。所有这一切都是在一般性的Kloosterman层附加到还原集团引入\cite{HNY}。作为应用，我们给出了R. Evans（\cite{Evans0}，\cite{Evans}）用显式模形式的傅里叶系数表示经典Kloosterman和的七次和八次对称幂矩.第八对称幂矩猜想的证明依赖于C. D. C. D.
Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals
设$K$是一个固定的数域，假设是$\mathbb Q$上的伽罗瓦。设$r$和$f$是固定整数，其中$f$为正。给定定义在$K$上的椭圆曲线$E$，考虑了Frobenius迹等于$r$的$K$的$f$素理想的次数的计数问题.除了在$f=2$的情况下，我们表明，“平均”的数量，这样的素理想的范数小于或等于$x$满足一个渐近恒等式，是按照标准的数学。这项工作涉及到经典的Lang-Trotter猜想，并扩展了几位作者的工作。
Statistics for traces of cyclic trigonal curves over finite fields
本文研究了q元域上亏格为g的循环三角曲线的Frobenius自同态的迹随曲线在模空间的不可约分量中的变化.我们证明了当q固定，g增加时，Frobenius迹的极限分布等于q+1个独立随机变量取0的概率为2/（q+2）和取1的概率为q/（3（q+2））的和，e^{（2 pi i）/3}，e^{（4pi i）/3}.这扩展了Kurlberg和Rudnick的工作，他们认为超椭圆曲线具有相同的极限。我们还证明了当g和q都趋于无穷大时，归一化迹具有标准的复高斯分布，以及如何将这些结果推广到投影线的p-折叠覆盖。
Using the Kuznetsov trace formula, we prove a spectral decomposition for the sums of generalized Dirichlet $L$-functions. Among applications are an explicit formula relating norms of prime geodesics to moments of symmetric square $L$-functions and an asymptotic expansion for the average of central values of generalized Dirichlet $L$-functions.
Bounds for the distribution of the Frobenius traces associated to a generic abelian variety
本文在广义黎曼假设下，研究阿贝尔簇的弗罗贝尼乌斯迹分布，得到关于素数分布的渐进行为，并在附加假设下进一步改进了这些界。
Traces of Hecke operators in level 1 and Gaussian hypergeometric functions
本文给出了第一层p阶Hecke算子的迹在有限域2F 1-超几何函数值下的计算公式，将作者以前的工作推广到所有奇素数p，而不仅仅是p=1（mod 12）的奇素数.首先给出了椭圆曲线族上Frobenius迹的一般一级迹公式，然后利用Lennon最近的工作给出了p >3的所有素数的超几何函数的一级迹公式.
Frobenius in $W(E_8)$ for $y^2=x^3+x+D(t)$ over $\mathbb F_q(t)$: ranks, Tate--Shafarevich groups and trace statistics
研究椭圆曲线在函数域F_q(t)上的鸣鸣现象，通过振荡的平均弗罗贝尼乌斯迹区分秩0和秩1曲线，揭示了BSD不变量与L多项式因子化的关系。
Statistics for biquadratic covers of the projective line over finite fields
研究了当曲线在模空间的不可约分支中变化时，作为$\mathbb{P}^{1}_{\mathbb{F}_{q}}$的四次非循环覆盖的亏格$g$曲线的Frobenius自同态的迹的分布.证明了当$q$固定时，Frobenius迹的极限分布等于$q + 1$独立随机离散变量之和.我们还表明，当这两个$g$和$q$去无穷大，规范化的轨迹有一个标准的复高斯分布。最后，我们将这些计算推广到一般情况下的任意覆盖的$\mathbb{P}^{1}_{\mathbb{F}_{q}}$与伽罗瓦群同构的$r$副本的$\mathbb{Z}/2\mathbb{Z}$。对于$r = 1$，我们恢复了已知的超椭圆情况。我们还包括一个附录，由Alina Bucur给出这些分布的启发式。
Distribution of the traces of Frobenius on elliptic curves over function fields
设C是定义在特征p>3的q元有限域$\mathbb{F}_{q}$上的光滑不可约射影曲线，$K=\mathbb{F}_{q}(C)$是它的函数域，$ϕ_{\mathcal{E}}:\mathcal{E}\to C$是$\mathbf{E}/K$的极小正则模型.对于每个$P\in C$，表示为$\mathcal{E}_P=ϕ^{-1}_{\mathcal{E}}(P)$。椭圆曲线$E/K$在$P\in C$上有良好的约简当且仅当$\mathcal{E}_P$是定义在$P$的剩余域$κ_P$上的椭圆曲线。这个域是$\mathbb{F}_q$的次数为$°(P)$的有限扩张。设$t(\mathcal{E}_P)=q^{°(P)}+1-#\mathcal{E}_P(κ_P)$是Frobenius在P处的迹。[10，第五章，定理2.4]），则$t(\mathcal{E}_P)$是$\mathcal{E}_P$的zeta函数的零点的逆的和。特别是$|t(\mathcal{E}_P)|\le 2q^{°(P)}$。设$C_0\subset C$是C中$E/K$具有良好约化的点集，$C_0(\mathbb{F}_{q^k})$是$C_0$的$\mathbb{F}_{q^k}$ -有理点的子集.我们讨论以下问题。设$k\ge 1$和t是整数，并假设$|t|\le 2q^{k/2}$。设$π(k,t)=…
Frobenius distribution for pairs of elliptic curves and exceptional isogenies
设E和E'是数域上的两条椭圆曲线。我们证明了E和E'在有限位置p的约化对无穷多个p是几何同构的，并得出了超奇异素数存在的结论。这个结果是一个类似的分布Frobenius迹的已知结果的密度的Noether-Lefschetz轨迹在霍奇理论。证明依赖于模曲线上Hecke对应的动力学性质。
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…
Let q be an odd power of a prime p and let A/Fq be a supersingular abelian variety of dimension g. We show that if p>2g+1, then the characteristic polynomial of the q-Frobenius is an even polynomial. This generalizes the well-known result on the vanishing of the trace of the p-Frobenius when p>3 for supersingular elliptic curves over Fp.
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication. For a prime $p$ of good reduction for $E$, we write $\#E_p(\mathbb{F}_p) = p + 1 - a_p(E)$ for the number of $\mathbb{F}_p$-rational points of the reduction $E_p$ of $E$ modulo $p$. Under the Generalized Riemann Hypothesis (GRH), we study the primes $p$ for which the integer $|a_p(E)|$ is a prime. In particular, we prove the following results: (i) the number of primes $p < x$ for which $|a_p(E)|$ is a prime is bounded from above by $C_1(E) \frac{x}{(\log x)^2}$ for some constant $C_1(E)$; (ii) the number…
Frobenius trace fields of cohomologically rigid local systems
本文研究上同调刚性局部系统模p约化后Frobenius迹所在数域F_p的有界性问题，证明了在特定monodromy条件下或单点有界即可推出整体有界，并探讨其与强几何来源的关系。
We express the Frobenius-Hecke traces on the compactly supported cohomology of a Shimura variety of abelian type in terms of elliptic parts of stable Arthur-Selberg trace formulas for the endoscopic groups. This confirms predictions of Langlands and Kottwitz at primes where the level is hyperspecial.
A standard observation in algebraic geometry and number theory is that a ramified cover of an algebraic variety $\widetilde{X}\rightarrow X$ over a finite field $F_q$ furnishes the rational points $x\in X(F_q)$ with additional arithmetic structure: the Frobenius action on the fiber over $x$. For example, in the case of the Vieta cover of polynomials over $F_q$ this structure describes a polynomial's irreducible decomposition type. Furthermore, the distribution of these Frobenius actions is encoded in the cohomology of $\widetilde{X}$ via the Grothendieck-Lefschetz trace formula. This note pres…
Approximations of $SL(3,\mathbb{Z})$ Hecke-Maass $L$-Functions by short Dirichlet polynomials
本文通过短Dirichlet多项式近似SL(3,Z) Hecke-Maass L函数，利用Kuznetsov迹公式方法，展示了L函数倒数的平均近似性。
Distribution of Farey Fractions in Residue Classes and Lang--Trotter Conjectures on Average
证明了$T$阶Farey分数的集合，即集合$\{α/β\in \Q : \gcd(α, β) = 1, 1 \le α, β\le T\}$，对于任何固定的$\eps>0$，在$T \ge p^{1/2 +\eps}$提供的素数$p$模的剩余类中是均匀分布的.我们应用此获得的上界的郎-特罗特曲率的Frobenius迹和Frobenius字段“平均”的一个参数的家庭的椭圆曲线。
We describe an algorithm for computing, for all primes $p \leq X$, the mod-$p$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometric motive in time quasilinear in $X$. This combines the Beukers--Cohen--Mellit trace formula with average polynomial time techniques of Harvey et al.
Bilinear forms with trace functions over arbitrary sets, and applications to Sato-Tate
本文证明了丰饶层迹函数在任意子集支撑下的一般双线性形式非平凡上界，结合加性组合与Kloosterman层乘积和结果突破了Pólya-Vinogradov障碍，并推导出稀疏族中Kloosterman和与椭圆曲线Frobenius迹的Sato-Tate分布。
We give an expression for the trace of Frobenius for the family of curves \[ y^3 = x (x-1)(x-λ)(x-μ) \] over finite fields in terms of finite field hypergeometric functions.
The bias conjecture for elliptic curves over finite fields and Hurwitz class numbers in arithmetic progressions
本文研究了Frobenius迹位于算术级数的有限域椭圆曲线的二阶矩偏差猜想，证明偏差对正密度级数为正、对正密度级数为负，与经典猜想相反，并给出了相关矩量的显式公式。
The computation of the order of Frobenius action on the $\ell$-torsion is a part of Schoof-Elkies-Atkin algorithm for point counting on an elliptic curve $E$ over a finite field $\mathbb{F}_q$. The idea of Schoof's algorithm is to compute the trace of Frobenius $t$ modulo primes $\ell$ and restore it by the Chinese remainder theorem. Atkin's improvement consists of computing the order $r$ of the Frobenius action on $E[\ell]$ and of restricting the number $t \pmod{\ell}$ to enumerate by using the formula $t^2 \equiv q (ζ+ ζ^{-1})^2 \pmod{\ell}$. Here $ζ$ is a primitive $r$-th root of unity. In …
We extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized cubic moment of modular L-functions at the central point s = 1/2 and for prime moduli q. As an application, we exploit our recent result on the mollification of the fourth moment of Dirichlet L-functions to derive that for any pair $(ω_1,ω_2)$ of multiplicative characters modulo q, there is a positive proportion of $χ$ (mod q) such that $L(χ, 1/2 ), L(χω_1, 1/2 )$ and $L(χω_2, 1/2)$ are simultaneo…
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。
Rationality of trace and norm L-functions
对于给定的有限域k和整数r上的交换代数群上的l-adic层F，定义了F在G（k）中点t处的r阶局部范数L-函数，并证明了它的合理性.该函数给出了F在G（k_r）（其中k_r是k的r次扩张）上的范数为t的点上的局部Frobenius迹的和的信息。对于G的一维仿射直线或环面，这些和可以反过来被用来估计曲线上的有理点的数量或指数和的绝对值，这些指数和在一大群平移或同态下是不变的。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。