Power sums of Hecke's eigenvalues and application

Average values of quadratic Hecke character sums

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 $n$.

将$q$-级数用Hecke型双重和的构建块表示

Expressing $q$-series in terms of building blocks of Hecke-type double-sums

本文将Wang、Yee和Liu研究的双重和用两类Hecke型双重和构建块表示，确定其（模拟）模性，并将Andrews的$q$-超几何函数表示为混合模拟模形式。

无零点区域在Hecke特征值的均值与移位卷积和上的应用

Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues

本文假设Vinogradov-Korobov型无零点区域与广义Ramanujan-Petersson猜想，建立了$SL(n,\mathbb{Z})$上Hecke-Maass尖形式傅里叶系数短和的非平凡上界，并应用于移位和均值及系数变号的条件性结果。

On three third order mock theta functions and Hecke-type double sums

We obtain four Hecke-type double sums for three of Ramanujan's third order mock theta functions. We discuss how these four are related to the new mock theta functions of Andrews' work on $q$-orthogonal polynomials and Bringmann, Hikami, and Lovejoy's work on unified Witten-Reshetikhin-Turaev invariants of certain Seifert manifolds. We then prove identities between these new mock theta functions by first expressing them in terms of the universal mock theta function.

Exponential sums with Dirichlet coefficients of L-functions

Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new conditional results on exponential sums with Hecke eigenvalues and squares of Hecke eigenvalues over primes. We employ these estimates to improve our recent result on squares of Hecke eigenvalues at Piatetski-Shapiro primes under the Riemann Hypothesis for symmetric square L-functions for Hecke eigenforms.

不可约二次多项式上Hecke特征值和的抵消现象与质量等分布

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）。

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。

通过超几何特征和得到的Hecke算子的迹

Traces of Hecke Operators via Hypergeometric Character Sums

本文提出一种一致且具几何性的方法，利用有限域上的超几何特征和，给出了算术三角群相关情形下尖点形式空间上Hecke算子迹的显式公式，适用于椭圆模曲线和Shimura曲线，并可求出特征值。

Quadratic Hecke sums and mass equidistribution

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…

关于Ikeda提升的Hecke特征值

On Hecke eigenvalues of Ikeda lifts

本文证明Ikeda提升在素数处的Hecke特征值为正，并给出所有素数下的上下界，方法涉及互反多项式表达。

On sums of Hecke series in short itervals

We prove $\sum_{K-G}\le κ_j \le K+G}α_j H_j^3(1/2) \ll_εGK^{1+ε}$ for $K^ε\le G \le K$, where $α_j = |ρ_j(1)|^2(\cosh πκ_j)^{-1}$, and $ρ_j(1)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue $λ_j = κ_j^2 + 1/4$ to which the Hecke series $H_j(s)$ is attached. This result yields the new bound $H_j(1/2) \ll_εκ_j^{1/3+ε}$.

The quantitative distribution of Hecke eigenvalues of Maass forms

Let $f$ be a normalized Hecke-Maass cusp form of weight zero for the group $SL_2(\mathbb Z)$. This article presents several quantitative results about the distribution of Hecke eigenvalues of $f$. Applications to the $Ω_{\pm}$-results for the Hecke eigenvalues of $f$ and its symmetric square sym$^2(f)$ are also given.

惯性素数处虚二次域的局部预解式、广义高斯和与反分圆Hecke $L$-值的$p$-adic赋值

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。

On the asymptotics of the shifted sums of Hecke eigenvalue squares

The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for $X^{\frac{2}{3}+ε} < H <X^{1-ε},$ there are constants $B_{h}$ such that $$ \sum_{X\leq n \leq 2X} λ_{f}(n)^{2}λ_{f}(n+h)^{2}-B_{h}X=O_{f,A,ε}\big(X (\log X)^{-A}\big)$$ for all but $O_{f,A,ε}\big(H(\log X)^{-3A}\big)$ integers $h \in [1,H]$ where $\{λ_{f}(n)\}_{n\geq1}$ are normalized Hecke eigenvalues of a fixed holomorphic cusp form $f.$ Our method is based on the Hardy-Littlewood circle method. We divide the minor arcs into two parts $m_{1}$ and $m_{2}.$ In order to tr…

Oscillations of Hecke Eigenvalues at Primes

In this paper, we are interested in exploring the cancellation of Hecke eigenvalues twisted with an exponential sums whose amplitude is $\sqrt{n}$ at prime arguments.

Sums of Hecke eigenvalues over quadratic polynomials

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.

Large Hecke eigenvalues and an Omega result for non Saito--Kurokawa lifts

We prove a result on the distribution of Hecke eigenvalues, $μ_F(p^r)$ (for $r=1,2$ or $3$) of a non Saito--Kurokawa lift $F$ of degree $2$. As a consequence, we obtain an Omega result for the Hecke eigenvalues for such an $F$, which is the best possible in terms of orders of magnitude.

Computing systems of Hecke eigenvalues associated to Hilbert modular forms

We utilize effective algorithms for computing in the cohomology of a Shimura curve together with the Jacquet-Langlands correspondence to compute systems of Hecke eigenvalues associated to Hilbert modular forms over a totally real field.

比较新形式的海克 eigenvalues

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。

Arithmetic behaviour of Hecke eigenvalues of Siegel cusp forms of degree two

Let $F$ and $G$ be Siegel cusp forms for $\Sp_4(\Z)$ and weights $k_1, k_2$ respectively. Also let $F$ and $G$ be Hecke eigenforms lying in distinct eigen spaces. Further suppose that neither $F$ nor $G$ is a Saito-Kurokawa lift. In this article, we study simultaneous arithmetic behaviour of Hecke eigenvalues of these Hecke eigenforms.

Hecke eigenvalues and relations for Siegel Eisenstein series of arbitrary degree, level, and character

We evaluate the action of Hecke operators on Siegel Eisenstein series of arbitrary degree, level and character. For square-free level, we simultaneously diagonalise the space with respect to all the Hecke operators, computing the eigenvalues explicitly, and obtain a multiplicity-one result. For arbitrary level, we simultaneously diagonalise the space with respect to the Hecke operators attached to primes not dividing the level, again computing the eigenvalues explicitly.

Hecke特征值的中心极限定理

Central limit theorem for Hecke eigenvalues

本文在分裂单代数群的一般设定下，利用紧致李群不可约特征证明了Hecke特征值的中心极限定理。

On sums of Hecke-Maass eigenvalues squared over primes in short intervals

We prove a uniform estimate for sums of Hecke--Maass eigenvalues squared over primes in short intervals that can be regarded as an analogue of Hoheisel's classical prime number theorem for all real analytic cusp forms. Our argument is modelled after our treatment of Linnik's least prime number theorem for arithmetic progressions (Tata LN 72) and depends on recent works in the theory of automorphic representations. We stress that constants in the present work, including those implicit, are all universal and effectively computable, although we pay no particular attention to numerical precision. …

The number of Hecke eigenvalues of same signs

We give the best possible lower bounds in order of magnitude for the number of positive and negative Hecke eigenvalues. This improves upon a recent work of Kohnen, Lau & Shparlinski. Also, we study an analogous problem for short intervals.

Hecke eigenvalues and relations for degree 2 Siegel Eisenstein series

We evaluate the action of Hecke operators on Siegel Eisenstein series of degree 2, square-free level and arbitrary character, without using knowledge of their Fourier coefficients. From this we construct a basis of simultaneous eigenforms for the full Hecke algebra, and we compute their eigenvalues. As well, we obtain Hecke relations among the Eisenstein series. Using these Hecke relations in the case that $\stufe$ is square-free and the character is trivial, we generate a basis for the space of Eisenstein series.

Hecke eigenvalues of Siegel modular forms of "different weights"

Given two Siegel eigenforms of different weights, we determine explicit sets of Hecke eigenvalues for the two forms that must be distinct. In degree two, and under some additional conditions, we determine explicit sets of Fourier coefficients of the two forms that must be distinct.

Hecke特征值的整性及Hecke域的成长

Integrality of Hecke eigenvalues and the growth of Hecke fields

本文证明了Hilbert和Siegel模形式的Hecke特征值为代数整数，并应用该整性估计了具有非零中心L值的Hilbert尖形式Hecke域的成长，以及GL_{2d}(A_Q)上尖自守表示有理域的成长。

On the distribution of Hecke eigenvalues for cuspidal automorphic representations for GL(2)

Given a self-dual cuspidal automorphic representation for GL(2) over a number field, we establish the existence of an infinite number of Hecke eigenvalues that are greater than an explicit positive constant, and an infinite number of Hecke eigenvalues that are less than an explicit negative constant. This provides an answer to a question of Serre. We also consider analogous problems for cuspidal automorphic representations that are not self-dual.

Asymptotics for Hecke eigenvalues of automorphic forms on compact arithmetic quotients

In this paper, we describe the asymptotic distribution of Hecke eigenvalues in the Laplace eigenvalue aspect for certain families of Hecke-Maass forms on compact arithmetic quotients. Instead of relying on the trace formula, which was the primary tool in preceding studies on the subject, we use Fourier integral operator methods. This allows us to treat not only spherical, but also non-spherical Hecke-Maass forms with corresponding remainder estimates. Our asymptotic formulas are available for arbitrary simple and connected algebraic groups over number fields with cocompact arithmetic subgroups…

On the occurrence of Hecke eigenvalues in sectors

Let $π$ be a non-self-dual unitary cuspidal automorphic representation of non-solvable polyhedral type for GL(2) over a number field. We show that $π$ has a positive upper Dirichlet density of Hecke eigenvalues in any sector whose angle is at least 2.63 radians.

The distribution of the eigenvalues of Hecke operators

For each prime $p$, we determine the distribution of the $p^{th}$ Fourier coefficients of the Hecke eigenforms of large weight for the full modular group. As $p\to\infty$, this distribution tends to the Sato--Tate distribution.

Kloosterman sums and Hecke polynomials in characteristics 2 and 3

In this paper we give a modular interpretation of the $k$-th symmetric power $L$-function of the Kloosterman family of exponential sums in characteristics 2 and 3, and in the case of $p=2$ and $k$ odd give the precise 2-adic Newton polygon. We also give a $p$-adic modular interpretation of Dwork's unit root $L$-function of the Kloosterman family, and give the precise 2-adic Newton polygon when $k$ is odd. In a previous paper, we gave an estimate for the $q$-adic Newton polygon of the symmetric power $L$-function of the Kloosterman family when $p \geq 5$. We discuss how this restriction on prim…

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…

Computing Hecke eigenvalues below the cohomological dimension

Let G be a torsion-free finite-index subgroup of SL(n,Z) or GL(n,Z), and let d be the cohomological dimension of G. We present an algorithm to compute the eigenvalues of the Hecke operators on the integral cohomology of degree d-1 for n = 2, 3, and 4. In addition, we describe a modification of the modular symbol algorithm of Ash-Rudolph for computing Hecke eigenvalues for the integral cohomology of degree d.

关于平均最小负 Hecke 特征值

On the average least negative Hecke eigenvalue

本文研究经典 newforms 的 Hecke 特征值首次变号的平均值，区分负素数特征值与一般负特征值，并证明在 GRH 下其与最小二次非剩余均值相同。

关于Ikeda提升的Hecke特征值的可整除性

On divisibility of Hecke eigenvalues of Ikeda lifts

本文估计了Ikeda提升的Hecke特征值被固定正整数整除的素数密度，并研究了相关椭圆Hecke本征形式的傅里叶系数在算术级数中的分布。

Large sums of Hecke eigenvalues of holomorphic cusp forms

Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and let $\{λ_f(n)\}_{n\geq 1}$ be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of $λ_f(n)$, we investigate the range of $x$ (in terms of $k$) for which there are cancellations in the sum $S_f(x)=\sum_{n\leq x} λ_f(n)$. We first show that $S_f(x)=o(x\log x)$ implies that $λ_f(n)<0$ for some $n\leq x$. We also prove that $S_f(x)=o(x\log x)$ in the range $\log x/\log\log k\to \infty$ assuming the Riemann hypothesis for $L(s, f)$, and furthermore that this range is best …

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。

Congruences modulo prime powers of Hecke eigenvalues in level $1$

We continue the study of strong, weak, and $dc$-weak eigenforms introduced by Chen, Kiming, and Wiese. We completely determine all systems of Hecke eigenvalues of level $1$ modulo $128$, showing there are finitely many. This extends results of Hatada and can be considered as evidence for the more general conjecture formulated by the author together with Kiming and Wiese on finiteness of systems of Hecke eigenvalues modulo prime powers at any fixed level. We also discuss the finiteness of systems of Hecke eigenvalues of level $1$ modulo $9$, reducing the question to the finiteness of a single e…

On systems of Hecke eigenvalues in cohomology of certain subgroups of GL_n(F)

We show how there is a natural action on the cohomology groups attached to certain subgroups of GL_n(F) of the Hecke operators defined as elements in an adelic double coset algebra. Our main result is, that if a system of eigenvalues for Hecke operators occur in the cohomology groups with coefficients in certain modules, then by changing the groups the system also occur in the cohomology groups with coefficients in a 1-dimensional module.

On the occurrence of large positive Hecke eigenvalues for GL(2)

Let $π$ be a cuspidal automorphic representation for GL(2)/$\mathbb{Q}$ that is self-dual. In this Note we show that there exists a positive upper Dirichlet density of primes at which the associated Hecke eigenvalues of $π$ are larger than a specified positive constant.

Ikeda提升的Hecke特征值

Hecke Eigenvalues of Ikeda Lifts

本文通过辛群Hecke代数的球面映射，推导了Ikeda提升的Hecke特征值λ_F(p^r)的显式公式，证明其可表示为具有正首项和有限系数的p^{±1/2}多项式，并由此得出对所有充分大的素数p，特征值为正。

Computation of Hecke eigenvalues (mod $p$) via quaternions

In a 1987 letter, Serre proves that the systems of Hecke eigenvalues arising from mod $p$ modular forms (of fixed level $Γ(N)$ coprime to $p$, and any weight $k$) are the same as those arising from functions $Ω(N) \to \bar{\mathbb F}_p$, where $Ω(N)$ is some double quotient of $D^\times (\mathbb A_f)$ and $D$ is the unique quaternion algebra over $\mathbb Q$ ramified at $\{p,\infty\}$. We present an algorithm which then computes these Hecke eigenvalues on the quaternion side in a combinatorial manner.

Analytic evaluation of Hecke eigenvalues for classical modular forms

We propose a method for computing approximations to the Hecke eigenvalues of a classical modular eigenform $f$, based on the analytic evaluation of $f$ at points in the upper half plane. Our approach works with arbitrary precision, allows for a strict control of the error in the approximation, and outperforms current exact computation methods.

On Short Sums Involving Fourier Coefficients of Maass Forms

We study sums of Hecke eigenvalues of Hecke-Maass cusp forms for the group $\mathrm{SL}(n,\mathbb Z)$, with general $n\geq 3$, over certain short intervals under the assumption of the generalised Lindelöf hypothesis and a slightly stronger upper bound concerning the exponent towards the Ramanujan-Petersson conjecture that is currently known. In particular, in this case we evaluate the second moment of the sums in question asymptotically.

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.

On the Occurrence of Hecke Eigenvalues and a Lacunarity Question of Serre

Let πbe a unitary cuspidal automorphic representation for GL(n) over a number field. We establish upper bounds on the number of Hecke eigenvalues of πequal to a fixed complex number. For GL(2), we also determine upper bounds on the number of Hecke eigenvalues with absolute value equal to a fixed number γ; in the case γ=0, this answers a question of Serre. These bounds are then improved upon by restricting to non-dihedral representations. Finally, we obtain analogous bounds for a family of cuspidal automorphic representations for GL(3).

On the distribution of the divisor function and Hecke eigenvalues

We investigate the behavior of the divisor function in both short intervals and in arithmetic progressions. The latter problem was recently studied by É. Fouvry, S. Ganguly, E. Kowalski, and Ph. Michel. We prove a complementary result to their main theorem. We also show that in short intervals of certain lengths the divisor function has a Gaussian limiting distribution. The analogous problems for Hecke eigenvalues are also considered.

On Hecke eigenvalues at primes of the form $[g(n)]$

In this paper, we study the average of the Fourier coefficients of a holomorphic cusp form for the full modular group at primes of the form $[g(n)]$.

Bounds on the multiplicity of the Hecke eigenvalues

Fix an integer $N$ and a prime $p\nmid N$ where $p\geq 5$. We show that the number of newforms $f$ (up to a scalar multiple) of level $N$ and even weight $k$ such that $\mathcal{T}_p(f)=0$ is bounded independently of $k$, where $\mathcal{T}_p(f)$ is the Hecke operator.

Hecke 算子特征值的平均大小

On the average size of the eigenvalues of the Hecke operators

本文利用二次均值，从纵向与横向两种视角确定了作用于尖点模形式空间 $S_k(Γ_0(N))$ 的 Hecke 算子特征值平均大小。

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。

Kuznetsov's trace formula and the Hecke eigenvalues of Maass forms

We give an adelic treatment of the Kuznetsov trace formula as a relative trace formula on GL(2) over Q. The result is a variant which incorporates a Hecke eigenvalue in addition to two Fourier coefficients on the spectral side. We include a proof of a Weil bound for the generalized twisted Kloosterman sums which arise on the geometric side. As an application, we show that the Hecke eigenvalues of Maass forms at a fixed prime, when weighted as in the Kuznetsov formula, become equidistributed relative to the Sato-Tate measure in the limit as the level goes to infinity.

Non-vanishing and sign changes of Hecke eigenvalues for Siegel cusp forms of genus two (with an Appendix by E. Kowalski and A. Saha)

In this paper, we show that half of non-zero coefficients of the spinor zeta function of a Siegel cusp form of genus 2 are positive and half are negative.