Amir Akbary, Alia Hamieh
We establish the two-dimensional asymptotic distributions of the logarithm and logarithmic derivative of $L$-functions associated with a family of cubic Hecke characters. A crucial ingredient in the proof of our main result is an exponential decay estimate for the characteristic functions of the distributions.
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In this paper, we study moments of central values of cubic Hecke $L$-functions in $\mathbb{Q}(i)$, and establish quantitative non-vanishing result for those values.
Towards Keating-Snaith's conjecture for cubic Hecke $L$-functions over the Eisenstein field
本文基于Radziwill和Soundararajan的工作，在广义黎曼假设下通过扩展David和Güloğlu的三次Hecke L函数零点1级密度扭曲估计，得到艾森斯坦域上稀疏三次Hecke L函数族的Keating-Snaith猜想条件下界。
We study the first moments of central values of Hecke $L$-functions associated with quadratic, cubic and quartic symbols to prime moduli. This also enables us to obtain results on first moments of central values of certain families of cubic and quartic Dirichlet $L$-functions of prime moduli.
Non-vanishing for cubic Hecke $L$-functions
本文研究了三次海克勒L-函数在中心点处非消失性的无条件结果，通过 mollified 短时域方法，首次给出了三次族的渐近评估并建立了新的上界。
Quantum variance for cubic moment of Hecke--Maass cusp forms and Eisenstein series
本文分别给出了Hecke--Maass尖点形式和Eisenstein级数三次矩方差的上界，基于大筛不等式和Lindelöf均值型界，并建立了特殊点附近GL(2)相关L-函数四次矩和八次矩的精确上界。
Let $k=\mathbb{Q}(\sqrt{-3})$, and let $c\in \mathfrak{O}_k$ be a square free algebraic integer such that $c\equiv 1~({\rm mod}~{\langle9\rangle})$. Let $ζ_{k(c^{1/3})}(s)$ be the Dedekind zeta function of the cubic field $k(c^{1/3})$ and $ζ_k(s)$ be the Dedekind zeta function of $k$. For fixed real $σ>1/2$, we obtain asymptotic distribution functions $F_σ$ for the values of the logarithm and the logarithmic derivative of the Artin $L$-functions \begin{equation*} L_c(σ)= \frac{ζ_{k(c^{1/3})}(σ)}{ζ_k(σ)}, \end{equation*} as $c$ varies. Moreover, we express the characteristic function of $F_σ$ e…
In this paper, we evaluate a smoothed character sum involving $\sum_{m}\sum_{n}\leg {m}{n}$, with quadratic, cubic or quartic Hecke characters $\leg {m}{n}$, and the two sums over $m$ and $n$ are of comparable lengths.
Joint cubic moment of Eisenstein series and Hecke-Maass cusp forms
本文研究谱参数趋于无穷时自守形式的联合三次矩，分别给出 Eisenstein 级数对角情形的上界估计，以及非对角情形和 Hecke-Maass 尖点形式混合情形的渐近消失结果。
On the cubic Shimura lift to $PGL(3)$: Hecke correspondences
本文为 Hecke 对应建立新基本引理，证明 $PGL_3(F)$ 与 $SL_3(F)$ 三次覆盖的球 Hecke 代数同构并匹配分布，为新的整体 Shimura 提升及周期刻画奠定关键基础。
Hecke $L$-functions Away From The Central Line
本文研究了三次Hecke L-函数在临界条带内s=1/3处的第一矩，揭示了相位转变现象，并改进了误差项和二次项，同时推测更高阶Hecke L-函数的行为。
The cubic moment of Hecke--Maass cusp forms and moments of $L$-functions
该研究证明Hecke–Maass尖形式的光滑三次矩多项式衰减（随机波猜想新案例），通过估计中心$L$-值矩，得到$\rm GL(3)\times GL(2)$等$L$-函数的矩界与新次凸界。
First moment of central values of Hecke $L$-functions with Fixed Order Characters
本文在广义黎曼假设下，利用双重狄利克雷级数方法，研究了不同虚二次数域上Hecke L-函数中心值的平滑第一矩的渐进行为，并给出了二次族的渐进行为公式。
Extreme values of Euler-Kronecker constants of abelian cubic fields
本文在广义黎曼猜想下，研究三次阿贝尔域和二次域的Euler-Kronecker常数的极值，给出了相应结果。
The Distribution of Integers in a Totally Real Cubic Field
Hecke利用狄利克雷级数的Fourier展开研究了二次无理数的分数部分的分布。这种方法由Behnke和Ash-Friedberg推广，用于研究任意次全实数域中迹的全正整数个数的分布.当场为立方时，我们证明了加权丢番图和的渐近性态与单位群的结构有关。主项可以用Grössencharacter $L$ -函数表示。
We determine a formula for the average values of L-series associated to eigenforms at complex values.
We study the first and second mollified moments of central values of a quadratic family of Hecke $L$-functions of prime moduli to show that more than nine percent of the members of this family do not vanish at the central values.
Non-vanishing of Hecke L-values modulo p
在这篇文章中，我们遵循Hida的方法，建立一个类似的华盛顿定理的非零模p的Hecke L-值的CM域与反分圆扭曲。
p-adic Eisenstein-Kronecker series and non-critical values of p-adic Hecke L-functions of an Imaginary Quadratic Field when the conductor is divisible by p
当代数Hecke特征标的导体可被$p$整除时，我们将与类数为1的虚二次数域的代数Hecke特征标相关联的非临界特殊值$p$-adic $L$ -函数与称为$p$ -adic Eisenstein-Kronecker级数的$p$ -adic Coleman函数相关联.
On the arithmetic properties of complex values of Hecke-Mahler series
本文给出了Hecke-Mahler级数在乘法群$\GG_m^2(\CC)$的代数点上的复值对$\QQ$的代数依赖关系.我们的结果包含Loxton和van der Poorten，Mahler和Masser以前的定理。
Non vanishing of Central values of modular L-functions for Hecke eigenforms of level one
设F（z）是一个新的具有平凡特征标的权为2k的水平1的形式，并假定F（z）是所有Hecke算子的非零本征形.本文研究了F的扭模L-函数的中心值的非零性。
We obtain an asymptotic formula for the fourth moment of central values of a family of quadratic Hecke $L$-functions in the Gaussian field under the generalized Riemann hypothesis (GRH). We also establish lower bounds unconditionally and upper bounds under GRH for higher moments of the same family.
Joint value distribution of Hecke--Maass forms
本文提出 Hecke–Maass 尖点形式联合分布猜想，并通过证明两个联合矩的条件性结果，验证了正交尖点形式的统计独立性。
Large values of Hecke-Maass L-functions with prescribed argument
本文研究了在给定辐角的临界线上，附于Maass型的L函数的大值的存在性。所获得的结果依赖于Soundararajan开发和Hough进一步发展的共振方法。
We introduce a new trace formula of Kuznetsov type involving the central standard L-values and the Whittaker periods of cuspidal automorphic representations of PGL_n(Q) which are spherical at the archimedean place. As an application, we show a simultaneous non-vanishing of standard L-values at n-1 points on the critical strip for infinitely many Hecke-Maass cuspidal newforms.
Determination of $GL(3)$ Hecke-Maass forms from twisted central values
设$π_1$和$π_2$是$SL(3,\mathbb{Z})$的两个Hecke-Maass尖点形式，使得对所有本原特征标$χ$都有$$ L(\tfrac{1}{2},π_1\otimesχ)=L(\tfrac{1}{2},π_2\otimesχ). $$，然后证明了$π_1=π_2$ .
Large values of quadratic Hecke $L$-functions with prime-related moduli of the Gaussian field
本文在广义黎曼假设下，利用共振方法证明高斯域中与素数相关模的二次Hecke $L$-函数在中心点可取到任意大的值。
Algebraicity of critical Hecke $L$-values
本文综述了 Hecke L-函数临界值代数性的已有结果，并介绍文献[Kings-Sprang]中的新进展。
Stability of $p$-adic valuations of Hecke L-values
本文证明了 CM 域上代数 Hecke 特征在 $\ell$-进反循环扭曲族（$\ell\neq p$）中临界 L-值的 $p$-稳定性结果。
Ratios conjecture of cubic $L$-functions of prime moduli
本文在广义黎曼假设下，针对三次 Hecke L 函数提出比率猜想，并应用该猜想计算了中心值的第一矩和一级密度。
Deligne's conjecture on the critical values of Hecke $L$-functions
本文通过将Kings-Sprang的Eisenstein-Kronecker类解释为Blasius反射动机的de Rham类，证明了任意代数Hecke特征下Deligne关于L-函数临界值的猜想，推广了Blasius仅在CM域成立的结果。
We evaluate the first three moments of central values of a family of qudratic Hecke $L$-functions in the Gaussian field with power saving error terms. In particular, we obtain asymptotic formulas for the first two moments with error terms of size $O(X^{1/2+\varepsilon})$. We also study the first and second mollified moments of the same family of $L$-functions to show that at least $87.5\%$ of the members of this family have non-vanishing central values.
Hecke operators on period functions for $Γ_0(n)$
Hecke算子在经典全纯尖点形式和相应的周期多项式上的矩阵表示是众所周知的。本文给出了同余子群$Γ_0(n)$的向量值周期函数的Hecke算子的表示。为此，我们使用从向量值尖点形式空间到向量值周期函数空间的积分变换。
Cyclotomic fields are generated by cyclotomic Hecke {\it L}-values of totally real fields, II
本文研究全实域上单个临界分圆 Hecke L 值能否生成相应分圆 Hecke 域，将 Jun-Lee-Sun 的平凡驯顺特征标结果推广到可解全实域上的非平凡 Hecke 特征标。
Split-CM points and central values of Hecke L-series
分裂CM点是模空间h_2/Sp_4（Z）中对应于椭圆曲线与相同复乘的乘积$E \times E'$的点。本文证明了给定类h_2/Sp_4（Z）中的分裂CM点的个数与Eichler研究的权3/2模形式的系数有关。这一结果的主要应用是给出了某个Hecke L-级数的中心值$L(ψ_N, 1)$的一个公式。Hecke特征标$ψ_N$是Gross研究的椭圆Q曲线A的规范Hecke特征标$ψ$的扭曲，并且$L(ψ, 1)$的公式以及推广由Villegas和Zagier证明。计算$L(\psin, 1)$的公式易于计算，并给出了数值例子。
Connections between Double Zeta Values relative to $μ_N$, Hecke Operators $T_N$, and Newforms of Level $Γ_0(N)$ for $N=2,3$
本文将研究与$μ_N$有关的双zeta值、Hecke算子$T_N$和$N=2,3$的新形式的水平$Γ_0(N)$之间的各种联系。这些不同的联系概括了鲍马德和施内普斯的著名观点。本文还对$\mathcal{S}_k^{\textrm{new}}(Γ_0(2))^{\pm}$，$\mathcal{S}_k^{\textrm{new}}(Γ_0(3))^{\pm}$与某些周期多项式空间之间的Eichler-Shimura-Manin对应给出了一个猜想.
We study smoothed character sums involving $\sum_{m,n} ( \frac{m}{n} )_2$, where $( \frac{m}{n} )_2$ denotes the quadratic symbol in the Gaussian field. We extend previously known results to obtain asymptotic formulas for the sums considered to larger ranges of $m$ and $n$.
Let $K$ be an imaginary quadratic number field and let $L(s,ξ_{\ell})$ denote the Hecke $L$-function to an angular character $ξ_{\ell}$ with frequency $\ell$. We detect values of $\log |L(\tfrac12,ξ_{\ell})|$ with size at least $(\sqrt{2} + o_K(1))(\log X / \log \log X)^{1/2}$ along each dyadic range $X \leqslant \ell \leqslant 2X$. This result relies on the resonance method, which is applied for the first time to this family of $L$-functions, where the classification and extraction of diagonal terms depends on the geometry of the complex embedding of $K$.
In this paper, we study moments of central values of certain unitary families of Hecke $L$-functions of the Gaussian field, and establish quantitative non-vanishing result for the central values. We also establish an one level density result for the low-lying zeros of these families of Hecke $L$-functions.
In this paper, we study the moments of central values of Hecke $L$-functions associated with quadratic characters in $\mathbb{Q}(i)$ and $\mathbb{Q}(ω)$ with $ω= exp(2πi/3)$ and establish some quantitative non-vanishing result for these $L$-values.
In this paper, we evaluate explicitly certain quadratic Hecke Gauss sums of $\mathbb{Q}(ω), ω=\exp \left( \frac {2πi}{3}\right)$. As applications, we study the moments of central values of quadratic Hecke $L$-functions of $\mathbb{Q}(ω)$, and establish quantitative non-vanishing result for the $L$-values. We also establish an one level density result for the low-lying zeros of quadratic Hecke $L$-functions of $\mathbb{Q}(ω)$.
Twisted Hecke L-values and period polynomials
设$f_1,...,f_d$是$Γ_0(N)$上偶权$2k$的尖点型空间的正交基.令$L(f_i,s)$和$L(f_i,χ,s)$分别表示$f_i$的$L$ -函数和由Dirichlet特征标$χ$表示的其扭曲。在本文中，我们得到了具有$0<m,n<2k$和适当奇偶性的整数$m$和$n$的值$L(f_i,χ,m)\overline{L(f_i,n)}$的“迹公式”。在N=1或N=2的情况下，该公式为我们精确地计算Hecke本征形$f$的比值$L(f,χ,m)/L(f,n)$的值提供了一种方便的方法。
Set $K=\mathbb{Q}(i)$ and suppose that $c\in \mathbb{Z}[i]$ is a square-free algebraic integer with $c\equiv 1 \imod{\langle16\rangle}$. Let $L(s,χ_{c})$ denote the Hecke $L$-function associated with the quartic residue character modulo $c$. For $σ>1/2$, we prove an asymptotic distribution function $F_σ$ for the values of the logarithm of \begin{equation*} L_c(s)= L(s,χ_c)L(s,\overlineχ_{c}), \end{equation*} as $c$ varies. Moreover, the characteristic function of $F_σ$ is expressed explicitly as a product over the prime ideals of $\mathbb{Z}[i]$.
We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as $\mathrm{GSp}_2(\mathbb{A})$, $\mathrm{SO}(4,3)(\mathbb{\mathbb{A}})$ and $\mathrm{SO}(5,4)(\mathbb{A})$, where the prime modulus should, for various reasons, appear in the algebraic part of a critical "tensor-product" $L$-value associated to cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A})$ and $\mathrm{GSp}_2(\mathbb{A})$. Using special techniques for evaluating $L$-functions with few known coefficients, we compute sufficiently good approximations to detec…
In this paper, we prove the following asymptotic formula for the spectral cubic moment of central $L$-values: $$ \sum_{t_f \leqslant T} \frac {2 L \big( \tfrac 1 2 , f \big)^3} {L(1, \mathrm{Sym}^2 f)} + \frac {2} π \int_{0}^{T} \frac {\left| ζ\big(\tfrac 1 2 + it \big) \right|^{6} } { | ζ(1 + 2 it ) |^2 } \mathrm{d} t = T^2 P_3 (\log T) + O (T^{1+\varepsilon}) , $$ where $f$ ranges in an orthonormal basis of (even) Hecke--Maass cusp forms, and $P_3$ is a certain polynomial of degree $3$. It improves on the error term $O (T^{8/7+\varepsilon})$ in a paper of Ivić and hence confirms his strong c…
Let $p$ and $q$ be two distinct odd primes. Let $K$ be an imaginary quadratic field over which $p$ and $q$ are both split. Let $Ψ$ be a Hecke character over $K$ of infinity type $(k,j)$ with $0\le-j< k$. Under certain technical hypotheses, we show that for a Zariski dense set of finite-order characters $κ$ over $K$ which factor through the $\mathbb{Z}_q^2$-extension of $K$, the $p$-adic valuation of the algebraic part of the $L$-value $L(\overline{κΨ},k+j)$ is a constant independent of $κ$. In addition, when $j=0$ and certain technical hypothesis holds, this constant is zero.
On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters
该研究针对r≥3的Hecke L-函数，利用多重Dirichlet级数理论，推导其在含2r次单位根的整体域上的扭曲矩渐近公式，证明其中心值存在正比例非零值。
In this paper, we study the first moment of central values of Hecke $L$-functions associated with quartic characters.
Explicit hypergeometric modularity of certain weight two and four Hecke eigenforms
本文基于超几何模性方法，构造了两个eta商函数家族，利用Jacobi theta函数理论及其立方类比，将某些重量为二和四的海尔克埃钦形式的傅里叶系数表示为有限域周期函数，并获得新的模形式傅里叶系数与有限域Appell级数特殊值的关系式。
The $p$-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $L$-values of imaginary quadratic fields at inert primes
本文证明了虚二次域在惯性素数处沿反分圆塔的Hecke $L$-值 $p$-adic 赋值的渐近公式，通过确定广义高斯和与局部预解式的 $p$-adic 赋值，回答了 Rubin 的问题。
Distribution of central values of Hecke $L$-functions of $n$-th order characters
该研究针对n≥3的n阶Hecke特征稀疏族，证明了log|L(1/2,χ)|的单侧中心极限定理，结合相关方法与渐近公式，还得到0<k<1时2k阶矩的猜想量级上界。
A realization of the Hecke algebra on the space of period functions for Gamma_0(n)
Hecke代数在经典全纯尖点形式和相应的周期多项式上的标准实现是众所周知的。本文考虑Hecke代数在Hecke同余子群Gamma_0（n）的Maass尖点形式上的一个非标准实现.我们证明了最近由Hilgert，Mayer和Movasati导出的向量值周期函数作为Gamma_0（n）的转移算子的特殊本征函数确实与这些群的Maass尖点形式有关。这也导致了一个简单的解释的``Hecke喜欢''运营商的这些作者在上述非标准实现的Hecke代数空间的向量值周期函数。
Sums of Hecke eigenvalues along polynomial sequences and base change for $\text{GL}(2)$
研究GL(2)表示在所有有限处温顺时绝对值Hecke特征值之和，证明其仅当表示为cuspidal时才能超越平凡界。并将多项式值上的Hecke特征值之和问题与GL(2)的基变换问题联系起来。
Special values of symmetric power $L$-functions and Hecke eigenvalues
我们计算了模形式的对称幂的L-函数在临界带的边缘处的矩，该临界带被模形式的L-函数的中心值扭曲。我们表明，在偶数的权力的情况下，它是等价于扭曲的对称平方L-函数的临界带的边缘处的值。我们推导出对称功率L-函数的大小在临界带的条件下的边缘上的信息。在第二部分中，我们研究了小的和大的Hecke特征值的分布。我们推导出的同时极值条件的值的L-函数的对称权力的模形式的临界带的边缘上的信息。
Joint distribution of Hecke eigenforms on $\mathbb{H}^3$
本文研究了Hecke特征形式在H³上的联合分布，并支持正交尖点形式的统计独立性猜想。
Effective decorrelation of Hecke eigenforms
本文证明了权重趋于无穷大时两个Hecke特征形式值的有效定量去相关，并由此得出其线性组合的质量与零点的有效等分布版本。
Let $q:=e^{2 πiz}$, where $z \in \mathbb{H}$. For an even integer $k$, let $f(z):=q^h\prod_{m=1}^{\infty}(1-q^m)^{c(m)}$ be a meromorphic modular form of weight $k$ on $Γ_0(N)$. For a positive integer $m$, let $T_m$ be the $m$th Hecke operator and $D$ be a divisor of a modular curve with level $N$. Both subjects, the exponents $c(m)$ of a modular form and the distribution of the points in the support of $T_m. D$, have been widely investigated. When the level $N$ is one, Bruinier, Kohnen, and Ono obtained, in terms of the values of $j$-invariant function, identities between the exponents $c(m)$…
A note on $ L(1) $ of Hecke $ L-$series associated to the elliptic curves with CM by $ \sqrt{-3} $
考虑在二次域$\ \Q(\sqrt{-3}) $上定义的椭圆曲线$ E:\ y^{2} = x^{3} + D^{3} $。研究了附着在$ E $上的Hecke $ L-$级数，给出了它们在$ s=1, $上的值的计算公式和这些值的3-adic赋值的界.这些结果与[Q]和[QZ]中的结果是互补的，并且与Birch和Swinnerton-Dyer猜想的预言是一致的.
Generation of Hecke fields by squares of cyclotomic twists of modular $L$-values
本文证明非CM椭圆新形式及其分圆扭曲的Hecke域复合可由中心L值绝对值平方生成，方法基于单幂混合猜想解决技术。
We generalize a theorem by Titchmarsh about the mean value of Hardy's $Z$-function at the Gram points to the Hecke $L$-functions, which in turn implies the weak Gram law for them. Instead of proceeding analogously to Titchmarsh with an approximate functional equation we employ a different method using contour integration.
Special $L$-values of certain CM weight three Hecke eigenforms
本文通过超几何伽罗华表示的模性，将某些CM重量三的海克尔克雷恩形式的L-函数特殊值与拉曼纽金的替代基联系起来，并对相关表示进行完整分类。
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