On the moments of Hecke series at central points

Determination of $GL(3)$ Hecke-Maass forms from twisted central values

Suppose $π_1$ and $π_2$ are two Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ such that for all primitive character $χ$ we have $$ L(\tfrac{1}{2},π_1\otimesχ)=L(\tfrac{1}{2},π_2\otimesχ). $$ Then we show that $π_1=π_2$.

Moments and Non-vanishing of central values of Quadratic Hecke $L$-functions in the Gaussian Field

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.

The fifth moment of Hecke L-functions in the weight aspect

We prove an upper bound for the fifth moment of Hecke L-functions associated to holomorphic Hecke cusp forms of full level and weight k in a dyadic interval K < k < 2K, as K tends to infinity. The bound is sharp on Selberg's eigenvalue conjecture.

The first moment of twisted Hecke $L$-functions with unbounded shifts

We compute the first moment of twisted Hecke $L$-functions of prime power level going to infinity, uniformly in the conductor of the twist and in the vertical shift.

On the fourth moment of Hecke Maass forms and the Random Wave Conjecture

Conditionally on the Generalized Lindelöf Hypothesis, we obtain an asymptotic for the fourth moment of Hecke Maass cusp forms of large Laplacian eigenvalue for the full modular group. This lends support to the Random Wave Conjecture.

Hecke L-函数中心值的第一矩的初值

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。

Split-CM points and central values of Hecke L-series

Split-CM points are points of the moduli space h_2/Sp_4(Z) corresponding to products $E \times E'$ of elliptic curves with the same complex multiplication. We prove that the number of split-CM points in a given class of h_2/Sp_4(Z) is related to the coefficients of a weight 3/2 modular form studied by Eichler. The main application of this result is a formula for the central value $L(ψ_N, 1)$ of a certain Hecke L-series. The Hecke character $ψ_N$ is a twist of the canonical Hecke character $ψ$ for the elliptic Q-curve A studied by Gross, and formulas for $L(ψ, 1)$ as well as generalizations wer…

Moments and One level density of certain unitary families of Hecke {$L$}-functions

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.

Moments of Quadratic Hecke $L$-functions of Imaginary Quadratic Number Fields

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.

Sharp bound for the fourth moment of holomorphic Hecke cusp forms

We prove that the fourth moment of holomorphic Hecke cusp forms is bounded provided that the Riemann Hypothesis holds for an appropriate degree 8 L-function. We accomplish this using Watson's formula, which translates the question in hand into a moment problem for L-functions which is amenable to the techniques of Soundararajan and Harper on obtaining sharp bounds for moments of the Riemann zeta function.

First Moments of Some Hecke $L$-functions of Prime Moduli

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.

Moments and One level density of quadratic Hecke $L$-functions of $\mathbb{Q}(ω)$

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 $\mathbb{Q}(ω)$.

Hecke特征形式与L-函数混合矩的混合矩

Mixed moments of Hecke eigenforms and $L$-functions

研究Hecke特征形式不同重量的混合(2,2)矩的期望和方差估计，应用于三重积L-函数，基于L-函数的矩证明。

Hecke L-函数与黎曼ζ函数的混合积分矩

Mixed integral moments of the Hecke $L$-functions and Riemann zeta function

本文研究了Hecke L-函数与黎曼ζ函数在临界线上的混合矩，并建立了适用于解析形式和Maass形式的渐近公式。

n阶特征的Hecke L函数的中心值分布

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。

On the moments of Hecke series at central points II

We prove, in standard notation from spectral theory, the asymptotic formula ($B>0$) $$ \sum_{κ_j\le T}α_j H_j(1/2) = ({T\overπ})^2 - BT\log T + O(T(\log T)^{1/2}), $$ by using an approximate functional equation for $H_j(1/2)$ and the Bruggeman--Kuznetsov trace formula. We indicate how the error term may be improved to $O(T(\log T)^ε)$.

r阶特征的Hecke L-函数的二阶矩与中心值非零性

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。

The second moment of Dirichlet twists of Hecke $L$-functions

Fix a Hecke cusp form $f$, and consider the $L$-function of $f$ twisted by a primitive Dirichlet character. As we range over all primitive characters of a large modulus $q$, what is the average behavior of the square of the central value of this $L$-function? Stefanicki proved an asymptotic valid only for $q$ having very few prime factors, and we extend this to almost all $q$.

较短区间内全纯 Hecke 尖形式的四次矩

The fourth moment of holomorphic Hecke cusp forms in shorter intervals

本文研究了较短区间内全纯 Hecke 尖形式四次矩的平均值，通过设定区间长度 $H=K^{3/4+c}$，改进了 Khan 的先前结果。

二次海克尔L函数的第一矩与低阶项

First Moment of Quadratic Hecke $L$-Functions with Lower Order Term

本文在黎曼猜想和林德尔öf猜想下，利用双狄利克雷级数方法，研究了二次海克尔L函数在高斯域中的第一矩，获得带有次主要项和误差项的渐进行为。

黎曼ζ函数与Hecke $L$函数的积分矩

Integral moment of the Riemann zeta function and Hecke $L$ functions

本研究针对$SL(2,\mathbb{Z})$的Hecke尖形式$f$，推导了临界线上黎曼ζ函数与Hecke $L$函数乘积、以及$|\zeta^2(s)|$与$L(s,f)$乘积的混合矩渐近公式，推进了解析数论中$L$函数矩问题的研究。

六次 Hecke $L$-函数在 $\mathbb{Q}(\omega)$ 上的矩与单层密度

Moments and One level density of sextic Hecke $L$-functions of $\mathbb{Q}(ω)$

本文研究 $\mathbb{Q}(\omega)$ 上六次 Hecke $L$-函数的矩与低零点单层密度，并证明在 GRH 下至少 $2/45$ 的成员在中心点不消失。

赫克 - 马阿斯尖形式的谱第四矩

Spectral fourth moments of Hecke--Maaß cusp forms

研究\(\mathrm{GL}(2)\)自守形式谱矩，考虑谱参数在二进区间的赫克 - 马阿斯尖形式族于黑格纳点的第四矩，建立最优界并应用于移位卷积问题及逐点丢番图指数。

超几何矩与Hecke迹公式

Hypergeometric Moments and Hecke Trace Formulas

该研究推广了有限域上超几何函数矩的已有工作，利用新的超几何起源Hecke迹公式证明了Q上本原超几何数据的新情形结果，得到有限域代数公式并提出相关矩的猜想。

权方面 Hecke $L$-函数的第十二矩

The Twelfth Moment of Hecke $L$-Functions in the Weight Aspect

本文证明全纯 Hecke 尖点形式在权方面的 Hecke $L$-函数第十二矩上界，推出 Weyl 强度次凸性及几乎所有形式满足次 Weyl 界，关键依赖 Kuznetsov 谱互反公式。

Gaussian域中二次Hecke $L$-函数中心值的一阶矩

First moment of central values of quadratic Hecke $L$-functions in the Gaussian field

本文利用双重Dirichlet级数方法，计算了Gaussian域中二次Hecke $L$-函数中心值的平滑一阶矩，在广义Riemann假设下得到误差项为 $O(X^{1/4+\varepsilon})$ 的渐近公式，并推广至二重特征和情形。

On exponential sums with Hecke series at central points

Upper bound estimates for the exponential sum $$ \sum_{K<κ_j\le K'<2K} α_j H_j^3(1/2) \cos(\k_j\log({4{\rm e}T\over κ_j})) \qquad(T^ε\le K \le T^{1/2-ε}) $$ are considered, where $α_j = |ρ_j(1)|^2(\coshπκ_j)^{-1}$, and $ρ_j(1)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue $λ_j = κ_j^2 + {1\over4}$ to which the Hecke series $H_j(s)$ is attached. The problem is transformed to the estimation of a classical exponential sum involving the binary additive divisor problem. The analogous exponential sums with $H_j(\hf)$ or $H_j^2(\hf)$ replacing $H_j^3(1/2)$ a…

A Functional Equation for the Spectral Fourth Moment of Modular Hecke L-Functions

Albeit essential corrections are required both in his claim and in his argument, N.V. Kuznetsov observed in his Bombay article (*) of 1989 a highly interesting transformation formula for spectral sums of products of four values of modular Hecke L-functions. A complete proof of a corrected version of the formula was later supplied by the present author in 1991, which has, however, remained unpublished for more than a decade, except for a limited distribution. The aim of the present paper is to reproduce this proof of the author with some sophistications, retaining the original style as much as …

Hecke特征值和的二阶矩 II

The Second Moment of Sums of Hecke Eigenvalues II

本文研究了Hecke特征值和的二阶矩，揭示了在不同重量范围下矩的大小差异，为相关数论问题提供了新见解。

Corrigendum: The base change fundamental lemma for central elements in parahoric Hecke algebras

This note corrects a minor misstatement in section 2 of the paper in the title (arXiv:0808.3426). It also addresses some related issues. The error does not affect the main results of that paper, but nevertheless this corrigendum seems necessary to avoid potential confusion.

Hecke特征值和的二阶矩I

The Second Moment of Sums of Hecke Eigenvalues I

本文研究了Hecke特征值和的二阶矩，发现当求和长度x接近k²时，求和大小发生转变，且在后续工作中将进一步探讨x大于k²时的平均大小变化。

On effective determination of Maass forms from central values of Rankin-Selberg $L$-function

We address the problem of identifying a Hecke-Maass cusp form $f$ of full level from the central values of the Rankin-Selberg $L$-functions $L(1/2,f\otimes h)$ where $h$ runs through the set of Hecke-Maass eigenforms of full level. We prove a quantitative result in this direction.

On some conjectures and results for the Riemann zeta-function and Hecke series

Some results and conjectures on $Z_2(s) = \int_1^\infty |ζ(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are presented. Consequences of these conjectures regarding the eighth moment of $|ζ(1/2+it)$ and the error term in the fourth moment of $|ζ(1/2+it)$ are discussed.

A formula for the central value of certain Hecke L-functions

Let N = 1 mod 4 be the negative of a prime, K=Q(sqrt{N}) and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to 3 modulo 4. Under these assumptions, there exists Hecke characters $ψ_{\D}$ of K with conductor $(\D)$ and infinite type $(1,0)$. Their L-series L(ψ_\D,s)$ are associated to a CM elliptic curve E(N,\D) defined over the Hilbert class field of $K$. We will prove a Waldspurger-type formula for L(ψ_\D,s) of the form L(ψ_\D,1) = Ω\sum_{[\A],I} r(\D,[\A],I) m_{[\A],I}([\D]) where the sum is over class ideal representatives I of a maximal order in the quatern…

The base change fundamental lemma for central elements in parahoric Hecke algebras

Clozel and Labesse proved the base change fundamental lemma for spherical Hecke algebras attached to an unramified group over a p-adic field. This paper proves an analogous fundamental lemma for centers of parahoric Hecke algebras attached to the same class of groups. This provides an ingredient needed for the author's program to study Shimura varieties with parahoric level structure at p.

Hecke--Maass尖点形式和Eisenstein级数三次矩的量子方差

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。

高斯域中与素数相关模的二次Hecke $L$-函数的大值

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。

Central elements in affine mod $p$ Hecke algebras via perverse $\mathbb{F}_p$-sheaves

Let $G$ be a split connected reductive group over a finite field of characteristic $p > 2$ such that $G_\text{der}$ is absolutely almost simple. We give a geometric construction of perverse $\mathbb{F}_p$-sheaves on the Iwahori affine flag variety of $G$ which are central with respect to the convolution product. We deduce an explicit formula for an isomorphism from the spherical mod $p$ Hecke algebra to the center of the Iwahori mod $p$ Hecke algebra. We also give a formula for the central integral Bernstein elements in the Iwahori mod $p$ Hecke algebra. To accomplish these goals we construct …

Non-vanishing of the Central Derivative of Canonical Hecke L-functions

In the early 1980s, Rohrlich began a study of canonical Hecke characters, which are closely related to the simplest examples of CM elliptic curves. He and Montgomery showed the non-vanishing of the central value when the L-function has an even functional equation, and we now show the non-vanishing of the central derivative when the functional equation is odd. Using the results of Gross-Zagier and Kolyvagin-Logachev, we can apply the non-vanishing to the ranks and Shafarevitch-Tate groups of the Q-curves "A(p)" studied by Gross in his thesis. In particular, their rank is determined by a congrue…

Hecke-Maass cusp forms on PGL_n of large levels with non-vanishing L-values

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.

傅里叶谱互反性与典范赫克L-函数

Fourier Spectral Reciprocity and Canonical Hecke $L$-Functions

本文证明数域上的傅里叶型环面谱互反公式，将其应用于CM域的典范赫克L-函数，得到含中心值与导数的一致一阶矩公式，结合次凸性界给出相关定量非零及Mordell-Weil秩1结果。

Sup-norm of Hecke-Laplace Eigenforms on $S^3$

We prove sub-convex bounds on the fourth moment of Hecke-Laplace eigenforms on $S^3$. As a corollary, we get a bound on the sup-norm on an individual eigenform, which constitutes an improvement over what is achievable through employing the Iwaniec-Sarnak amplifier.

Simultaneous non-vanishing of central values of $GL(2) \times GL(3) $ and $GL(3)$ $L$-functions

We study simultaneous non-vanishing of $L(\tfrac{1}{2},\di)$ and $L(\tfrac{1}{2},g\otimes \di)$, when $\di$ runs over an orthogonal basis of the space of Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ and $g$ is a fixed $SL(2,\mathbb{Z})$ Hecke cusp form of weight $k\equiv 0 \pmod 4$.

高斯Hecke L函数L(s,λ^k)的移位二阶矩

Shifted second moment of Gaussian Hecke $L$-functions $L(s,λ^k)$

该研究建立了高斯角Hecke L函数L(s,λ^k)的光滑加权移位二阶矩的一致渐近公式，通过转化非对角贡献为Weyl和等方法完成证明，所得公式与L函数比率猜想的四交换预测一致，误差为O_{Φ,ε}(K^{1/2+ε})。

Gelfand--Graev表示作为GL(r)有限中心覆盖的单型Hecke代数模

Gelfand--Graev representation as a Hecke algebra module of simple types of a finite central cover of $\mathrm{GL}(r)$

针对满足gcd(n,p)=1的非阿基米德局部域上一般线性群的n重覆盖，将Gelfand--Graev表示实现为单型Hecke代数模并给出显式表达式，推导得离散系列表示的Whittaker维数，结合Zelevinsky分类可理论上得到所有不可约表示的Whittaker维数。

由具有非零扭曲中心 $L$-值的本原形式所张成的子空间

Subspaces spanned by eigenforms with nonvanishing twisted central $L$-values

本文构造了整权本原形式中非零二次扭曲中心 $L$-值子空间与半整权本原形式中特定非零 Fourier 系数子空间的显式生成集，并证明二者经 Shimura 提升同构。

Hecke特征值和的低阶矩

Low moments of Hecke eigenvalue sums

本文证明Sato--Tate随机乘性函数部分和的优于平方根相消，并借助乘法混沌与Harper去随机化方法，首次在GL(2)自守形式中导出Hecke特征值和及特征形式低阶矩的上界，引入Hecke s-范数。

二元二次型表示的整数处Hecke特征值的一阶矩

First moment of Hecke eigenvalues at the integers represented by binary quadratic forms

本文研究稀疏整数集索引的Hecke特征形式傅里叶系数求和估计，给出特定二元二次型表示的无平方因子整数对应的和式上界，并确定其Hecke特征值首次符号改变的量级，改进推广了已有结果。

Hecke特征形式的联合分布

Joint distribution of Hecke eigenforms

本文提出关于多个Hecke特征形式联合分布的猜想，证明了在广义黎曼猜想和广义拉马努金猜想下，两个Hecke特征形式联合质量的渐进行为，并展示了两个Hecke特征形式渐近去相关性消失的结论。

Non-vanishing modulo p of central critical Rankin-Selberg L-values with anticyclotomic twists

We prove non-vanishing modulo p, for a prime $\ell$ different from p, of central critical Rankin-Selberg L-values with anticyclotomic twists of $\ell$-power conductor. The L-function is Rankin product of a cusp form and a theta series of arithmetic Hecke character of an imaginary quadratic field. The paper is concerned with the case when the weight of Hecke character is greater than that of cusp form, so the L-value is essentially different in nature from the one in the landmark work of Vatsal and Cornut-Vatsal on the same theme.

On sum of Hecke eigenvalue squares over primes in very short intervals

Let $η>0$ be a fixed positive number, let $N$ be a sufficiently large number. In this paper, we study the second moment of the sum of Hecke eigenvalues over primes in short intervals (whose length is $η\log N$) on average (with some weights) over the family of weight $k$ holomorphic Hecke cusp forms. We also generalize the above result to Hecke-Maass cusp forms for $SL(2,\mathbb{Z})$ and $SL(3,\mathbb{Z}).$ By applying the Hardy-Littlewood prime 2-tuples conjecture, we calculate the exact values of the mean values.

Twisted moments of L-functions and spectral reciprocity

A reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q twisted by the ell-th Hecke eigenvalue as the fourth moment of automorphic L-functions of level ell twisted by the q-th Hecke eigenvalue. Direct corollaries include subconvexity bounds for L-functions in the level aspect and a short proof of an upper bound for the fifth moment of automorphic L-functions.

A higher moment formula for the Siegel--Veech transform over quotients by Hecke triangle groups

We compute higher moments of the Siegel--Veech transform over quotients of $SL(2,\mathbb{R})$ by the Hecke triangle groups. After fixing a normalization of the Haar measure on $SL(2,\mathbb{R})$ we use geometric results and linear algebra to create explicit integration formulas which give information about densities of $k$-tuples of vectors in discrete subsets of $\mathbb{R}^2$ which arise as orbits of Hecke triangle groups. This generalizes work of W.~Schmidt on the variance of the Siegel transform over $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.

GL(2) 与对称平方 L 函数的混合矩

Mixed moments of $\rm GL(2)$ and symmetric square $L$-functions

该研究推导短区间内 GL(2) 及其对称平方 L 函数混合矩的渐近公式，并据此定量证明两类自守形式对应 L 函数中心值的同时非零性。