Peng Gao
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.
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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.
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.
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。
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。
A Functional Equation for the Spectral Fourth Moment of Modular Hecke L-Functions
尽管在他的主张和他的论证中都需要进行必要的修正，库兹涅佐夫在1989年的孟买文章（*）中观察到了一个非常有趣的变换公式，用于模赫克L函数的四个值的乘积的谱和。1991年，本作者提供了该公式的修正版的完整证明，但该证明十多年来一直未发表，只有有限的发行。本文的目的是在尽可能保持原文风格的前提下，对作者的这一证明作一些说明。（*）他声称决议的第八时刻问题的黎曼ζ函数。但他的论点是致命的错误。我们的论文在这里涉及他的论点中的一个重要组成部分，但我们对八分之一时刻本身没有任何要求。
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.
We study real zeros of a family of quadratic Hecke $L$-functions in the Gaussian field to show that more than twenty percent of the members have no zeros on the interval $(0, 1]$.
Since the study by Jacobi and Hecke, Hecke-type series have received a lot of attention. Unlike such series associated with indefinite quadratic forms, identities on Hecke-type series associated with definite quadratic forms are quite rare in the literature. Motivated by the works of Liu, we first establish many parameterized identities with two parameters by employing different $q$-transformation formulas and then deduce various Hecke-type identities associated with definite quadratic forms by specializing the choice of these two parameters. As applications, we utilize some of these Hecke-typ…
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 $\mathbb{Q}(ω)$.
Murmurations of quadratic Hecke $L$-functions of the Gaussian field
研究高斯域二次赫克L函数族，在广义黎曼假设下评估其蜂群密度。
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.
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.
$p$-adic Beilinson conjecture for ordinary Hecke motives associated to imaginary quadratic fields
本文的目的是给出一系列的文件[BK 1]，[BK 2]的概述有关$p$ -adic Beilinson猜想的动机与Hecke字符的虚二次域$K$，一个素数$p$分裂在$K$ .
In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke $L$-functions of imaginary quadratic number fields of class number one. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27... \%$ of the $L$-functions under consideration do not vanish at $1/2$.
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。
On binary quadratic forms and the Hecke groups
本文给出了系数在Z[lambda]中的二元二次型的约化理论，其中lambda是Hecke群中的最小平移.我们将模群Gamma（1）= SL（2，Z）推广到Hecke群，并广泛使用了修正的负连分数.
In this paper, we prove some one level density results for the low-lying zeros of famliies of quadratic and quartic Hecke $L$-functions of the Gaussian field. As corollaries, we deduce that, respectively, at least $94.27 \%$ and $5\%$ of the members of the quadratic family and the quartic family do not vanish at the central point.
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.
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 $n$.
In this paper, we study the one level density of low-lying zeros of a family of quadratic Hecke $L$-functions to prime moduli over the Gaussian field under the generalized Riemann hypothesis (GRH) and the ratios conjecture. As a corollary, we deduce that at least $75 \%$ of the members of this family do not vanish at the central point under GRH.
Ratios conjecture for primitive quadratic Hecke $L$-functions
本文在广义黎曼假设下利用多重Dirichlet级数，对类数为1的虚二次域原始二次Hecke $L$-函数族建立了含单位移的比值猜想，并给出中心值一阶矩的无条件渐近公式。
We consider the analogue of the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke eigenvalues summed over quadratic progressions. Our reduction provides an analogue for the compact case of a criterion established by Luo--Sarnak for the case of the non-compact modular surface. The novelty is that known proofs of such criteria have depended crucially upon Fourier expansions, which are not available in the compact case. Unconditionally, we establish a twisted varian…
In this paper, we apply the ratio conjecture of $L$-functions to derive the lower order terms of the $1$-level density of the low-lying zeros of a family quadratic Hecke $L$-functions in the Gaussian field. Up to the first lower order term, we show that our result is consistent with that obtained from previous work under the generalized Riemann hypothesis, when the Fourier transforms of the test functions are supported in $(-2, 2)$.
Hecke's integral formula for quadratic extensions of a number field
设K/F是数域的二次扩张。在发展了F上的Eisenstein级数的理论之后，我们证明了一个公式，该公式将K的部分zeta函数表示为Eisenstein级数的某个积分。作为应用，我们得到了K和F的zeta函数在s=1处的0阶Laurent系数的Kronecker型极限公式.
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。
Ratios conjecture of quadratic Hecke $L$-functions of prime-related moduli
该研究在广义黎曼假设下，针对类数为1的虚二次数域中与素数相关模的二次狄利克雷及Hecke L-函数族，建立了带移位的L-函数比率猜想，并据此渐近计算了中心值一阶矩与单级密度。
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.
On Jannsen's conjecture for Hecke characters of imaginary quadratic fields
我们提出了一个收集的结果Jannsen猜想的$p$ -进实现相关的Hecke字符在虚二次域$K$的类数1。对于纯局部型的伽罗瓦群，这个猜想很容易检验。在虚二次域$K$在$p$上的几何正则性条件下证明了这个猜想，它与整体伽罗瓦群是纯局部型的性质有关.如果没有这种规律性假设在$p$，我们提出了一个审查的关键情况下，在非关键的情况下实现相关的赫克字符在$K$。我们把这个猜想与Hecke特征标的相关$p$ -adic $L$ -函数的某些具体非临界值的非零性联系起来。最后，我们证明了该猜想遵循岩泽理论的一般猜想，几乎所有的泰特扭曲。
Weighted fourth moments of Hecke zeta functions with groessencharacters
我们利用最近得到的Kloosterman和的界来限制和$\sum_{-D\leq d\leq D} \int_{-D}^D |ζ(1/2+it,λ^d)|^4| \sum_{0<|μ|^2\leq M} A(μ)λ^d((μ)) |μ|^{-2it}|^2 {\rm d}t$，其中$λ^d$是满足$λ^d((α)) = λ^d(α{\Bbb Z}[i]) = (α/|α|)^{4d}$的Groessen特征，对于$0\neqα\in{\Bbb Z}[i]$，$ζ(s,λ^d)$是满足$\Re(s)>1$的Hecke zeta函数，而$D,M\in(0,\infty)$和函数$A:{\Bbb Z}[i]-\{0\}\rightarrow{\Bbb C}$是任意的（尽管只有在$M$相对于$D$较小的情况下，我们的结果才是新的和有趣的）。我们的一个新的界限可能有一个应用，使一定的改进的结果的P.A.刘易斯关于高斯素数分布的研究。
In this paper, we study lower order terms of the $1$-level density of low-lying zeros of quadratic Hecke L-functions in the Gaussian field. Assuming the Generalized Riemann Hypothesis, our result is valid for even test functions whose Fourier transforms are supported in $(-2, 2)$. Moreover, we apply the ratios conjecture of $L$-functions to derive these lower order terms as well. Up to the first lower order term, we show that our results is consistent with each other, when the Fourier transforms of the test functions are supported in $(-2, 2)$.
On Bloch-Kato's Tamagawa number conjecture for Hecke characters of imaginary quadratic number fields
这基本上是作者提交给芝加哥大学的论文（1997年5月）。证明了Bloch-Kato的Tamagawa数猜想对某些Hecke特征的有效性。利用Kato的显式互反律，研究了一维Lubin-Tate形式群的指数映射和奇素数（包括普通素数和超奇素数）的局部Tamagawa数.我还研究了P-部分的Shafarevich-Tate群的动机与Hecke字符，使用鲁宾的主要猜想在岩泽理论。有趣的是，椭圆曲线的p-adic周期与权为1的Eisenstein级数在挠CM点处的一个同余性质在Bloch-Kato猜想的证明中起着至关重要的作用.
An Eisenstein ideal for imaginary quadratic fields and the Bloch-Kato conjecture for Hecke characters
对于虚二次域F的某些代数Hecke特征，定义了p进Hecke代数中作用于GL_2/F的尖点自守形式的Eisenstein理想.通过寻找Eisenstein上同调类（G. Harder）和尖点类，证明了Hecke代数中Eisenstein理想的指数的一个下界.我们进一步证明了它的指数由与chi^{-1}相关联的p-adic Galois特征标的Selmer群的阶从上有界。它使用了R的工作。Taylor等人关于将伽罗瓦表示附加到GL_2/F的尖形上的问题。这些结果一起意味着一个下界的大小的塞尔默群的L（0，chi），符合由布洛赫-加藤猜想给出的值。
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.
Ratios conjecture for quadratic Hecke $L$-functions in the Gaussian field
本文在广义黎曼假设下，利用多重Dirichlet级数建立了高斯域二次Hecke $L$-函数族的比值猜想，并得到中心值一阶矩的渐近公式及误差项$O(X^{1/2+\varepsilon})$。
On some conjectures and results for the Riemann zeta-function and Hecke series
给出了关于$Z_2(s) = \int_1^\infty |ζ(1/2+ix)|^4x^{-s}dx (\Re s > 1)$的一些结果和图解。讨论了有关$|ζ(1/2+it)$的八阶矩和$|ζ(1/2+it)$的四阶矩中的误差项的结果。
Cancellation in Sums of Hecke Eigenvalues Over Quadratic Polynomials and Mass Equidistribution
研究短区间不可约二次多项式上Hecke特征值和的抵消现象，通过Hecke形式基平均得到平方根抵消，结合Kloosterman和界及Nelson工作，证明几乎所有模形式满足质量等分布猜想且收敛速度为k^(-δ)（δ<1/2）。
Murmurations of Hecke $L$-Functions of Imaginary Quadratic Fields
该研究计算了虚二次域非平凡特征Hecke $L$-函数族的murmuration密度，验证其渐近符合Katz-Sarnak 1级密度猜想，发现该族murmuration具有独特的近周期特性。
The critical order of certain Hecke L-functions of imaginary quadratic fields
令$-D < -4$表示一个基本判别式，该判别式是奇数或可被8整除，因此存在$\Bbb Q(\sqrt{-D})$的规范Hecke特征标。设$d$是$D$的一个基本判别素数。令$2k-1$是与$\Bbb Q(\sqrt{-D})$的类数互素的奇数自然整数。令$χ$是$\Bbb Q(\sqrt{-D})$的规范Hecke特征标的$(2k-1)$次幂乘以Kronecker符号$n\mapsto(\frac{d}{n})$的扭曲。本文证明了Hecke $L$ -函数$L(s,χ)$在其中心点$s=k$的阶由其根数决定，当$|d| \leq c(ε)D^{{1/24}-ε}$或当$|d| \leq c(ε)D^{\frac1{12} -ε}$和$k\geq 2$时，其中$ε> 0$和$c(ε)$是只依赖于$ε$的常数.
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。
Köhler's Conjecture on Hecke Theta Series of Weight One
通过从二次域特征诱导的二维Galois表示推导，证明了Günter Köhler关于Hecke theta级数中二次域歧义的猜想。
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。
Sums of Hecke eigenvalues over quadratic polynomials
设f（z）= sum_n a（n）n^{（k-1）/2} e（nz）是Gamma 0（N）的一个尖点形式，特征标chi和权kgeq为4.设q（x）= x^2 + sx + t是整系数多项式。证明了对于依赖于f和q的常数c，sum_{n \leq X} a（q（n））= cX + O（X^{6/7+eps}）.在许多情况下，常数为零，例如，如果k是偶数。在此基础上，导出了半整数权和不同符号项的库兹涅佐夫公式。
Weyl's law for Hecke operators on GL(n) over imaginary quadratic number fields
本文证明了虚二次数域上Hecke算子作用于GL（n）的尖点形式的迹的一个中值定理以及误差项的一个上界，该误差项与Hecke算子有关.
Hecke operators and Hilbert modular forms
设F是实二次域，其整数环为O，类数为类数1.设Gamma是GL_2（O）的同余子群.我们描述了一种计算Hecke算子在上同调H^3（Gamma; C）上作用的方法。对于F实二次这个上同调群包含尖点上同调对应尖点希尔伯特模形式的平行重量2。因此，这种技术提供了一种方法来计算这些希尔伯特模形式上的Hecke作用。
The behavior of Hecke's L-function of real quadratic fields at s=0
对于一族实二次域$\{K_n=\FQ(\sqrt{f(n)})\}_{n\in \FN}$，一个Dirichlet特征标$χ$模$q$和一个预定理想$\{\fb_n\subset K_n\}$，研究了部分Hecke L函数$L_{K_n}(s,χ_n:=χ\circ N_{K_n},\fb_n)$的特殊值在$s=0$上的线性行为.我们表明，$n=qk+r$，$L_{K_n}(0,χ_n,\fb_n)$可以写为$$\frac{1}{12q^2}(A_χ(r)+kB_χ(r)),$$其中$A_χ(r),B_χ(r)\in \FZ[χ(1),χ(2),..., χ(q)]$，如果在$\fb_n$的连续分数方面的一定条件得到满足。此外，我们使用伯努利多项式的值精确地写出$A_χ(r)$和$B_χ(r)$。我们描述了如何使用线性解决一类数问题的一些家庭，并恢复在某些情况下的证明。最后，我们列出了几类具有线性的实二次域。
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 quadratic large sieve inequality over number fields
给出并证明了数域上二次特征标的一个大筛法不等式。为此，我们引入了n阶Hecke族的概念。我们发展这些Hecke家庭的基本理论，包括版本的泊松求和公式。
We make use of Hecke operators and arithmetic of imaginary quadratic fields to derive an explicit version of a special case of Siegel's mass formula.
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。
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。
Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields
本文给出了实二次域上Hilbert模形式的Hecke界和Sturm界的一个类比。设$K$是实二次域，$\Om_K$是其整数环.设$Γ$是$\SL_2(\Om_K)$的同余子群，$M_{(k_1,k_2)}(Γ)$是$Γ$的权$(k_1,k_2)$的Hilbert模形式空间.第一个主要结果是一个算法来构建一个有限集$S$，根据$K$，$Γ$和$(k_1,k_2)$，这样，如果傅立叶展开系数的形式$G \in M_{(k_1,k_2)}(Γ)$消失的集合$S$，那么$G$是零形式。第二个结果对应于Sturm情形中的相同陈述，即假设所有形式为$G$的Fourier系数都位于$\Q$的有限扩张中，并且设$\id{p}$是该扩张中的素理想，其范数在$K$中是非分歧的;进一步假设$G$的傅立叶展开系数对于$S$中的所有元素都位于理想的$\id{p}$中，则它们都位于理想的$\id{p}$中。
Hecke operators, Hecke Eigensystems, and Formal Modular Forms over Number Fields
本文提出了一种数字段上形式模形式的显式理论，通过模点定义Hecke算子和特征系统，并展示了如何利用主Hecke算子计算自守形式。
In this paper we prove the weak local Tamagawa number conjecture for the remaining non-critical cases for the motives associated to Hecke characters $ψ_θ:\mathbb{A}_K\to K^*$ of the author's previous paper, where $K$ is an imaginary quadratic field with $cl(K)=1$, under certain restrictions which originate mainly from the Iwasawa theory of imaginary quadratic fields.
We study the fourth moment of quadratic Dirichlet $L$-functions at $s= \frac{1}{2}$. We show an asymptotic formula under the generalized Riemann hypothesis, and obtain a precise lower bound unconditionally. The proofs of these results follow closely arguments of Soundararajan and Young [19] and Soundararajan [17].
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。
Local constancy of dimensions of Hecke eigenspaces of Automorphic forms
我们使用Buzzard的方法来研究不同类型的模形式-古典，在虚二次域和全实域的p进家庭。在偶次全实域的情形下，得到了定斜率变权空间维数的局部不变性。对于虚二次域，我们得到了与这种空间的维数上的权无关的界。
We study the $1$- or $2$-level density of families of $L$-functions for Hecke--Maass forms over an imaginary quadratic field $F$. For test functions whose Fourier transform is supported in $\left(-\frac 32, \frac 32\right)$, we prove that the $1$-level density for Hecke--Maass forms over $F$ of square-free level $\mathfrak{q}$, as $\mathrm{N}(\mathfrak{q})$ tends to infinity, agrees with that of the orthogonal random matrix ensembles. For Hecke--Maass forms over $F$ of full level, we prove similar statements for the $1$- and $2$-level densities, as the Laplace eigenvalues tends to infinity.
The fourth moment of quadratic Dirichlet $L$-Functions II
本研究无条件证明二次Dirichlet $L$-函数四次矩的含四个主项的渐近公式，基于已有工作引入改进大筛估计、交叉抵消分析等新要素，采用精细组合论证完成证明。
Arithmetic of Hecke L-functions of quadratic extensions of totally real fields
本文解决了 Shintani 方法在高维全实域上的无效性，为窄类数为1的全实域中导子为惰性有理素数的特征提供了 Shintani sets 的组合描述，简化了相关 Hecke L-函数，并将组合类数公式推广至全实域的二次扩张，有效肯定了 Hecke 猜想。
Let $K=\mathbb{Q}(\sqrt{-D})$ be an imaginary number field, $(p)=\mathfrak{p}\mathfrak{p}'$ be a split odd prime and $ψ$ be a Hecke character of conductor $\mathfrak{p}$. Let $L(s,ψ)$ be the associated $L$-function. We prove the Burgess bound in $t$-aspect and a hybrid bound in conductor aspect, \begin{equation*} L(1/2+it,ψ)\ll_{D,\varepsilon} (1+|t|)^{3/8+\varepsilon}p^{1/8} \end{equation*} for $p\ll t$. In Appendix A, we present the ideas for an elementary proof of Voronoi summation formula for holomorphic cusp forms with CM and squarefree level. This is done by exploiting the lattice struct…
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