Peng Gao
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 $(0, 1]$.
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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.
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.
We relate non-critical special values $p$-adic $L$-functions associated to algebraic Hecke characters of an imaginary quadratic number field with class number one to $p$-adic Coleman function called the $p$-adic Eisenstein-Kronecker series, when the conductors of the algebraic Hecke characters are divisible by $p$.
The purpose of this article is to give an overview of the series of papers [BK1], [BK2] concerning the $p$-adic Beilinson conjecture of motives associated to Hecke characters of an imaginary quadratic field $K$, for a prime $p$ which splits in $K$.
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 $1/2$.
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.
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。
We present a theory of reduction of binary quadratic forms with coefficients in Z[lambda], where lambda is the minimal translation in a Hecke group. We generalize from the modular group Gamma(1) = SL(2,Z) to the Hecke groups and make extensive use of modified negative continued fractions.
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 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 $(-2, 2)$.
Let K/F be a quadratic extension of number fields. After developing a theory of the Eisenstein series over F, we prove a formula which expresses a partial zeta function of K as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker's type which relates the 0-th Laurent coefficients at s=1 of zeta functions of K and F.
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.
We present a collection of results on a conjecture of Jannsen about the $p$-adic realizations associated to Hecke characters over an imaginary quadratic field $K$ of class number 1. The conjecture is easy to check for Galois groups purely of local type. We prove the conjecture under a geometric regularity condition for the imaginary quadratic field $K$ at $p$, which is related to the property that a global Galois group is purely of local type. Without this regularity assumption at $p$, we present a review of the known situations in the critical case and in the non-critical case for the realiza…
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 $(-2, 2)$.
This is essentially the author's thesis submited to The University of Chicago (May 1997). I prove the validity of Tamagawa number conjecture of Bloch-Kato for certain Hecke characters. I study the exponential map and local Tamagawa number for all odd primes (both ordinary and supersingular), using Kato's explicit reciprocity law for one dimensional Lubin-Tate formal group. I also study p-part of Shafarevich-Tate group for motives associated to Hecke characters, using Rubin's Main Conjecture in Iwasawa theory. Interestingly a congruence property between p-adic periods of elliptic curve and weig…
For certain algebraic Hecke characters chi of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL_2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-value L(0,chi). We further prove that its index is bounded from above by the order of the Selmer group of the p-adic Galois character associated to chi^{-1}. This uses the work of R. Taylor et al. on a…
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$O(X^{1/2+\varepsilon})$。
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）。
We prove a mean value theorem for the traces of Hecke operators acting on cusp forms of GL(n) over imaginary quadratic number fields together with an upper bound for the error term depending explicitly on the Hecke operator.
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。
Let $-D < -4$ denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of $\Bbb Q(\sqrt{-D})$ exists. Let $d$ be a fundamental discriminant prime to $D$. Let $2k-1$ be an odd natural integer prime to the class number of $\Bbb Q(\sqrt{-D})$. Let $χ$ be the twist of the $(2k-1)$th power of a canonical Hecke character of $\Bbb Q(\sqrt{-D})$ by the Kronecker's symbol $n\mapsto(\frac{d}{n})$. It is proved that the order of the Hecke $L$-function $L(s,χ)$ at its central point $s=k$ is determined by its root number when $|d| \leq c(ε)D^{{1/24}-ε}$…
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。
Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \leq X} a(q(n)) = cX + O(X^{6/7+eps}) for some constant c depending on f and q. The constant vanishes in many cases, for example if k is even. On the way a Kuznetsov formula for half-integral weight and entries having different sign is derived.
Let F be a real quadratic field with ring of integers O and with class number 1. Let Gamma be a congruence subgroup of GL_2 (O). We describe a technique to compute the action of the Hecke operators on the cohomology H^3 (Gamma; C). For F real quadratic this cohomology group contains the cuspidal cohomology corresponding to cuspidal Hilbert modular forms of parallel weight 2. Hence this technique gives a way to compute the Hecke action on these Hilbert modular forms.
For a family of real quadratic fields $\{K_n=\FQ(\sqrt{f(n)})\}_{n\in \FN}$, a Dirichlet character $χ$ modulo $q$ and prescribed ideals $\{\fb_n\subset K_n\}$, we investigate the linear behaviour of the special value of partial Hecke's L-function $L_{K_n}(s,χ_n:=χ\circ N_{K_n},\fb_n)$ at $s=0$. We show that for $n=qk+r$, $L_{K_n}(0,χ_n,\fb_n)$ can be written as $$\frac{1}{12q^2}(A_χ(r)+kB_χ(r)),$$ where $A_χ(r),B_χ(r)\in \FZ[χ(1),χ(2),..., χ(q)]$ if a certain condition on $\fb_n$ in terms of its continued fraction is satisfied. Furthermore, we write precisely $A_χ(r)$ and $B_χ(r)$ using values…
We formulate and prove a large sieve inequality for quadratic characters over a number field. To do this, we introduce the notion of an n-th order Hecke family. We develop the basic theory of these Hecke families, including versions of the Poisson summation formula.
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。
In this article we give an analogue of Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields. Let $K$ be a real quadratic field and $\Om_K$ its ring of integers. Let $Γ$ be a congruence subgroup of $\SL_2(\Om_K)$ and $M_{(k_1,k_2)}(Γ)$ the space of Hilbert modular forms of weight $(k_1,k_2)$ for $Γ$. The first main result is an algorithm to construct a finite set $S$, depending on $K$, $Γ$ and $(k_1,k_2)$, such that if the Fourier expansion coefficients of a form $G \in M_{(k_1,k_2)}(Γ)$ vanish on the set $S$, then $G$ is the zero form. The second result corresponds to th…
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.
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。
We use a method of Buzzard to study p-adic families of different types of modular forms - classical, over imaginary quadratic fields and totally real fields. In the case of totally real fields of even degree, we get local constancy of dimensions of spaces of fixed slope and varying weight. For imaginary quadratic fields we obtain bounds independent of the weight on the dimensions of such spaces.
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.
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…
The $1$-Level Density for Zeros of Hecke $L$-Functions of Imaginary Quadratic Number Fields of Class Number $1$
本文研究类数为1的虚二次数域上与角特征相关的Hecke L-函数族，利用Ratios Conjecture计算其零点平均1-水平密度的条件渐近公式（含低阶项），并证明支集在(-1,1)内的无条件结果与预测一致。
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.
This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity.
We show that Landau-Siegel zeros for Dirichlet $L$-functions do not exist or quantum unique ergodicity for $\mathrm{GL}_2$ Hecke-Maass newforms holds with an effective rate of convergence. This follows from a more general result: Landau-Siegel zeros of Dirichlet $L$-functions repel the zeros of all other automorphic $L$-functions from the line $\mathrm{Re}(s)=1$.
Extreme values of quadratic Hecke $L$-functions
本文在GRH下研究二次Hecke L函数在导子方面的极值，证明特定点处存在大值，并给出达到较小常数的特征数量下界，同时推广到函数域并给出固定条带及s=1处的界。
Quadratic forms of modular forms
研究全纯尖形式空间中的二次型，条件性地证明当两个Hecke特征形式的二次型没有公共对角项时，其内积收敛于公共非对角系数乘积之和，并定义ℓ^p范数建立条件上界，揭示Hecke基下二次型系数的不均匀小性。
In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. We observe that the functions that appear in Ramanujan's identities can be obtained from a Hecke action on a certain family of eta products. We establish further Hecke-type relations for these functions involving binary quadratic forms. Our observations enable us to find new identities for the Rogers-Ramanujan functions and also to use such identities in turn to find identities involving binary quadratic forms.
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.
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。
We prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon of Deuring and Heilbronn for Hecke $L$-functions. In forthcoming work of the second author, these estimates will be used to establish explicit bounds on the least norm of a prime ideal in a congruence class group and improve upon existing explicit bounds for the least norm of a prime ideal in the Chebotarev density theorem.
On $L$-functions of Hecke characters and anticyclotomic towers
本文推广 Rohrlich 的结果，证明虚二次域上一类反循环扭转 Hecke 特征的 $L$-函数在中心点的零点阶数除有限例外外由根数决定，为 0 或 1。
Comparing Hecke eigenvalues of newforms
研究两个不同新形式的海克 eigenvalues 在素数上的主导密度，证明至少为1/16，并在无复乘和非二次扭情况下进一步证明其平方的类似结果。
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