The cubic moment of central values of automorphic L-functions

The shifted fourth moment of automorphic L-functions of prime power level

We prove the asymptotic formula for the fourth moment of automorphic $L$-functions of level $p^ν$, where $p$ is a fixed prime number and $ν\rightarrow \infty$. This paper is a continuation of work by Rouymi, who computed asymptotics of the first three moments at prime power level, and a generalization of results obtained for prime level by Duke, Friedlander & Iwaniec and Kowalski, Michel & Vanderkam.

Automorphic properties of generating functions for generalized odd rank moments and odd Durfee symbols

We define two-parameter generalizations of Andrews' $(k+1)$-marked odd Durfee symbols and $2k$th symmetrized odd rank moments, and study the automorphic properties of some of their generating functions. When $k=0$ we obtain families of modular forms and mock modular forms. When $k \geq 1$, we find quasimodular forms and quasimock modular forms.

The sixth moment of automorphic $L$-functions

We investigate the sixth moment of the family of $L$-functions associated to holomorphic modular forms on $GL_2$ with respect to a congruence subgroup $Γ_1(q)$. We improve on previous work and obtain an unconditional upper bound of the correct order of magnitude.

The eighth moment of the family of $Γ_1(q)$-automorphic $L$-functions

We prove a Lindelöf on average bound for the eighth moment of a family of $L$-functions attached to automorphic forms on $GL(2)$, the first time this has been accomplished. Previously, such a bound had been proven for the sixth moment for our family by Djanković and for a similar family by Young. Our proof rests on a new approach which overcomes the lack of perfect orthogonality in the family initially observed by Iwaniec and Xiaoqing Li.

Integral Moments of Automorphic L-functions

This paper exposes the underlying mechanism for obtaining second integral moments of $GL_2$ automorphic $L$--functions over an arbitrary number field. Here, moments for $GL_2$ are presented in a form enabling application of the structure of adele groups and their representation theory. To the best of our knowledge, this is the first formulation of integral moments in adele-group-theoretic terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers $\Bbb{Q}$, we recover the classical results.

A note on the second moment of automorphic L-functions

We obtain the formula for the twisted harmonic second moment of the $L$-functions associated with primitive Hecke eigenforms of weight 2. A consequence of our mean value theorem is reminiscent of recent results of Conrey and Young on the reciprocity formula for the twisted second moment of Dirichlet $L$-functions.

自守L-函数矩的下界

Lower Bounds for Moments of Automorphic L-functions

本文提出一种方法，为所有正实数k，获得中心点处L-函数2k次矩的下界，并应用于GL_2和GL_3上自对偶尖峰自守表示的二次扭族。

自守形式的混合四阶矩与 $L$-函数的移位矩

Mixed fourth moments of automorphic forms and the shifted moments of $L$-functions

本文在GRH和GRC假设下，研究自守形式与Eisentein级数的混合四阶矩，揭示非均匀分布现象并提出相关猜想。

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.

First moments of Rankin-Selberg convolution of Automorphic Forms on $GL(2)$

We obtain a first moment formula for Rankin-Selberg convolution $L$-series of holomorphic modular forms or Maass forms of arbitrary level on $GL(2)$, with an orthonormal basis of Maass forms. One consequence is the best result to date, uniform in level, spectral value and weight, for the equality of two Maass or holomorphic cusp forms if their Rankin-Selberg convolutions with the orthonormal basis of Mass forms $u_j$ is equal at the center of the critical strip for sufficiently many $u_j$. The main novelty of our approach is the new way the error terms are treated. They are brought into an exa…

Second Moments and simultaneous non-vanishing of GL(2) automorphic L-series

We obtain a second moment formula for the L-series of holomorphic cusp forms, averaged over twists by Dirichlet characters modulo a fixed conductor Q. The estimate obtained has no restrictions on Q, with an error term that has a close to optimal power savings in the exponent. However, one of the contributions to the main term is a special value of a shifted double Dirichlet series. We show that this special value is small on average, and obtain a corresponding estimate for a mean value of the second moment over Q. This mean value is non-zero even when applied to a product of two distinct L-ser…

Subconvexity bounds in depth-aspect for automorphic L-functions on GL(2)

From a spectral identity we obtain asymptotics with error term for the second integral moments of families of automorphic L-functions for GL(2) over an arbitrary number field according to twists by idele characters with arbitrary ramification at a fixed finite place. The power-saving in the error term breaks convexity at this non-archimedean place.

Eisenstein series and the cubic moment for PGL(2)

Following a strategy suggested by Michel--Venkatesh, we study the cubic moment of automorphic $L$-functions on $\operatorname{PGL}_2$ using regularized diagonal periods of products of Eisenstein series. Our main innovation is to produce vectors whose integral transforms achieve arbitrarily weighted moments. Applications include general Motohashi-type identities and Weyl-type subconvex bounds for some families of $L$-functions, extending some results of Conrey--Iwaniec and Petrow--Young to the number field setting. We deduce improved estimates for representation numbers of ternary quadratic for…

大型正交族自守$L$-函数的$n^{th}$中心矩

The $n^{th}$ centered moments of a large orthogonal family of automorphic $L$-functions

本文研究了与level $q$全纯Hecke newforms相关联的大型正交族$L$-函数的one level densities的第$n$次中心矩，验证了Katz-Sarnak猜想，并解决了幻影超大项和组合问题。

Motohashi's fourth moment identity for non-archimedean test functions and applications

Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic L-functions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet L-functions modulo q weighted by a non-archimedean test function. This establishes a new reciprocity formula. As an application, we obtain sharp upper bounds for the fourth moment twisted by the square of a Dirichlet polynomial of length q^{1/4}. An auxiliary result of independent interest is a sharp upper …

三次层面中三重积 L-函数的 Weyl 界

The Weyl bound for triple product L-functions in the cubic level

本文在三次层面方面研究三重积 L-函数的强次凸性界，通过结合 Petersson 迹公式、Voronoi 求和、Jutila 圆法、Kuznetsov 迹公式和谱大筛不等式等工具证明了 Weyl 型界。

Automorphic properties of generating functions for generalized rank moments and Durfee symbols

We define two-parameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We prove that three specializations of the associated generating functions are so-called quasimock theta functions, while a fourth specialization gives quasimodular forms. We then define a two-parameter generalization of Andrews' smallest parts function and note that this leads to quasimock theta functions as well. The automorphic properties are deduced using q-series identities relating the relevant generating functions to known mock theta func…

Triple Shifted Sums of Automorphic L-Functions

In this work we provide a meromorphic continuation in three complex variables of two types of triple shifted convolution sums of Fourier coefficients of holomorphic cusp forms. The foundations of this construction are based in the continuation of the spectral expansion of a special truncated Poincaré series recently developed by Jeffrey Hoffstein. As a result we are able to produce previously unstudied and nontrivial asymptotics of truncated shifted sums which we expect to correspond to off-diagonal terms in the third moment of automorphic L-functions.

关于到PGL(3)的三次Shimura提升：基本引理

On the Cubic Shimura lift to $PGL(3)$: The Fundamental Lemma

本文研究了从SL₃三重覆盖到PGL₃的Shimura提升，并通过全球周期非消失性刻画提升像，从而建立基本引理。

自守随机乘法函数和的低阶矩

Low moments of automorphic random multiplicative function sums

本文确定了满足素平方消去条件的自守随机乘法函数部分和低阶矩的阶，推广了Harper的结果，适用于酉型和辛型不可约紧群表示。

Uniform subconvexity and symmetry breaking reciprocity

A non-symmetric reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q and primitive central character twisted by the l-th Hecke eigenvalue as a twisted mixed moment of automorphic L-functions of level l and trivial central character. As an application, uniform subconvexity bounds for L-functions in the level and the eigenvalue aspect are derived.

$θ= \infty$ 猜想与自守 $L$-函数的黎曼假设

The $θ= \infty$ Conjecture and the Riemann Hypothesis for Automorphic $L$-functions

本文研究自守 $L$-函数的 $θ=\infty$ 猜想，证明若其磨光二阶矩对任意多项式长度磨光器有界，则 $L$-函数在临界带相应区域非零，且该猜想对函数族蕴含准黎曼假设。

Co-periods and central symmetric cube L-values

In this article, we study the co-period integral attached to an automorphic form on $\GL(2)$ and two exceptional theta series on the cubic Kazhdan-Patterson cover of $\GL(2)$. In the local aspect, we show the $\Hom$-space is always of one dimension and conduct the unramified calculations. In the global aspect, we give the Euler decomposition for the co-period integrals of Eisenstein series and propose an Ichino-Ikeda type conjecture relating the co-period integrals of cuspidal forms to the central critical value of symmetric cube $L$-functions. We also deduce from the local multiplicity one re…

三次 twists 的 $GL_n(\mathbb{Q})$ $L$-函数非消失

Non-Vanishing of Cubic Twists of $GL_n(\mathbb{Q})$ $L$-functions

本文研究了三次Dirichlet字符与$GL_n(\mathbb{Q})$自动表示的$L$-函数非消失问题，证明了在特定实部条件下存在无限多个原始三次Dirichlet字符使$L$-函数不为零。

A uniform asymptotic formula for the second moment of primitive L-functions on the critical line

We prove an asymptotic formula for the second moment of automorphic L-functions of even weight and prime power level. The error term is estimated uniformly in all parameters: level, weight, shift and twist.

L-函数的宽矩 I：虚二次域类群特征的扭曲

Wide moments of $L$-functions I: Twists by class group characters of imaginary quadratic fields

本文通过推导一般权 Waldspurger 公式并结合 Heegner 点等分布性，计算了虚二次域 Hecke 特征扭曲的 Rankin--Selberg L-函数中心值的宽矩，并由此得到弱同时非零结果及三重卷积 L-函数非零性的新方法。

Riemann zeta 函数与 Dirichlet $L$-函数的混合矩

Mixed Moments of the Riemann Zeta and Dirichlet $L$-Functions

本文证明 Riemann zeta 函数与本原 Dirichlet 特征对应 $L$-函数的混合二阶矩 Motohashi 公式，并通过自守形式中心 $L$-值给出三次矩侧表达。

一个大型正交自守 L-函数族的低矮零点

Low-lying zeros of a large orthogonal family of automorphic $L$-functions

本文研究与级 q 的全纯 Hecke 新形式相关的新正交 L-函数族，证明其单层密度结果并将检验函数 Fourier 变换支集扩展至 (−4, 4)，为高阶矩、零点分布和临界线定理等后续研究奠定基础。

Moments of central values of cubic Hecke $L$-functions of $\mathbb{Q}(i)$

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.

Cubic moments of Fourier coefficients and pairs of diagonal quartic forms

We establish the non-singular Hasse Principle for pairs of diagonal quartic equations in 22 or more variables. Our methods involve the estimation of a certain entangled two-dimensional 21st moment of quartic smooth Weyl sums via a novel cubic moment of Fourier coefficients.

三次和四次狄利克雷L函数的位移矩

Shifted moments of cubic and quartic Dirichlet $L$-functions

本文在广义黎曼猜想下，建立了三次和四次狄利克雷L函数位移矩的上界，并应用此结果给出了三次和四次狄利克雷特征和的矩界。

Periods and Reciprocity I

Given $\mathbf{F}$ a number field with ring of integers $\mathcal{O}_{\mathbf{F}}$ and $\mathfrak{p},\mathfrak{q}$ two squarefree and coprime ideals of $\mathcal{O}_{\mathbf{F}}$, we prove a reciprocity relation for the first moment of the triple product $L$-functions $L(π\otimesπ_1\otimesπ_2,\frac{1}{2})$ twisted by $λ_π(\mathfrak{p})$, where $π_1$ and $π_2$ are a fixed unitary automorphic representation of $\mathrm{PGL}_2(\mathbb{A}_{\mathbf{F}})$ with $π_1$ cuspidal and $π$ runs through unitary automorphic representations of conductor dividing $\mathfrak{q}$. The method uses adelic integral…

$L$-函数二次扭的负一阶矩

Negative first moment of quadratic twists of $L$-functions

该研究在广义黎曼假设与拉马努金-彼得松猜想下，利用多重Dirichlet级数渐近求解了自守$L$-函数二次扭族在大于1/2点处的负一阶矩，推进了数论中$L$-函数矩的相关研究。

Bounds for moments of cubic and quartic Dirichlet $L$-functions

We study the $2k$-th moment of central values of the family of primitive cubic and quartic Dirichlet $L$-functions. We establish sharp lower bounds for all real $k \geq 1/2$ unconditionally for the cubic case and under the Lindelöf hypothesis for the quartic case. We also establish sharp lower bounds for all real $0 \leq k<1/2$ and sharp upper bounds for all real $k \geq 0$ for both the cubic and quartic cases under the generalized Riemann hypothesis (GRH). As an application of our results, we establish quantitative non-vanishing results for the corresponding $L$-values.

$GL_2$ $L$-函数三次矩的渐近公式

An asymptotic formula for a cubic moment of $GL_2$ $L$-functions

本文推导了自对偶GL2 L-函数三次矩的渐近公式，包含猜想主项和依赖于额外平均量的幂次节省误差项。

Mollification of the Fourth Moment of Dirichlet L-functions

We evaluate some twisted fourth moment of Dirichlet $L$-functions at the central point s=1/2 and for prime moduli q. The principal tool is a careful analysis of a shifted convolution problem involving the divisor function using spectral theory of automorphic forms and bounds for bilinear forms in Kloosterman sums. Having in mind simultaneous non vanishing results, we apply the Theorem to establish an asymptotic formula of a mollified fourth moment for this family of $L$-functions

Proof of the Strong Ivić Conjecture for the Cubic Moment of Maass-form $L$-functions

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…

函数域上三次L-函数平移值矩的上界

Upper bound for the moment of shifted values of cubic $L$-functions over function fields

该研究针对函数域上三次L-函数的平移值相关性，推导了其矩的上界，结果适用于q≡2 mod 3的非库默情形，q≡1 mod 3的库默情形可类似处理。

三重乘积$L$-函数一阶矩的谱互反性及其应用

Spectral Reciprocity for the first moment of triple product $L$-functions and applications

本文研究数域上三重乘积$L$-函数的一阶矩，通过谱方法得到扭曲一阶矩的谱互反公式，并应用于水平方面的次凸问题，给出相应次凸界。

艾森斯坦域上三次狄利克雷 L-函数的加扰矩

Mollified Moments of Cubic Dirichlet L-Functions over the Eisenstein Field

在广义黎曼假设下，证明艾森斯坦域上三次狄利克雷 L-函数在中心点有正密度不消失，通过计算第一加扰矩并估计高阶矩上界，不消失比例显式但极小。

A generalized cubic moment and the Petersson formula for newforms

Using a cubic moment, we prove a Weyl-type subconvexity bound for the quadratic twists of a holomorphic newform of square-free level, trivial nebentypus, and arbitrary even weight. This generalizes work of Conrey and Iwaniec in that the newform that is being twisted may have arbitrary square-free level, and also that the quadratic character may have even conductor. One of the new tools developed in this paper is a more general Petersson formula for newforms of square-free level.

关于 L-函数混合矩中的误差项

On the error term in a mixed moment of L-functions

本文研究了 Dirichlet L-函数第四矩及被 Dirichlet 特征扭曲的自守 L-函数混合矩渐近式中的误差项优化问题，并成功改进了后者的误差项。

模形式L函数在重量方面的移位第四矩

The shifted fourth moment of modular form L-functions in the weight aspect

本文研究了模形式L函数在重量方面的移位第四矩，并将其与黎曼ζ函数和自守L函数的矩联系起来，探讨了主项结构以推广到更高矩。

本原三次L-函数的扭曲二阶矩

Twisted second moment of primitive cubic L-functions

本文研究非Kummer设定下$\mathbb{F}_q(T)$上本原三次L-函数扭曲二阶矩的均值，采用双重Dirichlet级数方法建立了相应的误差项。

Algebraicity of the central critical values of twisted triple product $L$-functions

We study the algebraicity of the central critical values of twisted triple product $L$-functions associated to motivic Hilbert cusp forms over a totally real étale cubic algebra in the totally unbalanced case. The algebraicity is expressed in terms of the cohomological period constructed via the theory of coherent cohomology on quaternionic Shimura varieties developed by Harris. As an application, we generalize our previous result on Deligne's conjecture for certain automorphic $L$-functions for ${\rm GL}_3 \times {\rm GL}_2$. We also establish a relation for the cohomological periods under tw…

On a cubic moment of Hardy's function with a shift

An asymptotic formula for $$ \int_{T/2}^{T}Z^2(t)Z(t+U)\,dt\qquad(0< U = U(T) \le T^{1/2-\varepsilon}) $$ is derived, where $$ Z(t) := ζ(1/2+it){\bigl(χ(1/2+it)\bigr)}^{-1/2}\quad(t\in\Bbb R), \quad ζ(s) = χ(s)ζ(1-s) $$ is Hardy's function. The cubic moment of $Z(t)$ is also discussed, and a mean value result is presented which supports the author's conjecture that $$ \int_1^TZ^3(t)\,dt \;=\;O_\varepsilon(T^{3/4+\varepsilon}). $$

$L$-函数立方矩的指定局部分量家族的 Lindelöf 平均上界

The cubic moment of $L$-functions for specified local component families

本文研究了 $L$-函数中心值的立方矩在特定局部分量家族上的 Lindelöf 平均上界，通过新发展的 Petersson/Bruggeman-Kuznetsov 公式，推导了 Weyl 强子凸界，尤其在平方满和深度方面。

一大类自守L-函数的低阶零点密度研究

On the Density of Low Lying Zeros of a Large Family of Automorphic $L$-functions

该研究在GRH下推广了相关技术，改进了自守L-函数中心矩的密度结果，扩展了Katz-Sarnak密度预测的成立范围，相关技术可用于多素数求和量的研究。

单基因化三次域的2-类群大小的二阶矩

The second moment of the size of the $2$-class group of monogenized cubic fields

本文证明按高度排序的单基因化三次数域中，全实与复情形的2-类群大小二阶矩分别至多为3和6，且全实窄2-类群二阶矩至多为9，为单基因性改变类群分布提供了新证据。

三次狄利克雷L-函数在函数域上的二阶矩

The second moment of cubic Dirichlet L-functions over function fields

本文在有理函数域上研究三次狄利克雷L-函数中心点的二阶矩，推广了先前的一阶矩结果，利用解析技巧并关联高斯和平均值，获得渐近公式。

GL(2) 自守形式的代数扭曲

Algebraic twists of GL(2) automorphic forms

本文研究 GL2 自守形式 Hecke 特征值与有限域上迹函数的相关性，利用 adele 语言和相对迹公式，将 Burgess 型幂次改进结果推广至数域情形。

Level Reciprocity in the twisted second moment of Rankin-Selberg L-functions

We prove an exact formula for the second moment of Rankin-Selberg $L$-functions $L(1/2,f \times g)$ twisted by $λ_f(p)$, where $g$ is a fixed holomorphic cusp form and $f$ is summed over automorphic forms of a given level $q$. The formula is a reciprocity relation that exchanges the twist parameter $p$ and the level $q$. The method involves the Bruggeman/Kuznetsov trace formula on both ends; finally the reciprocity relation is established by an identity of sums of Kloosterman sums.

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。

The second moment of $\mathrm{GL}(n)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions

We prove an asymptotic expansion of the second moment of the central values of the $\mathrm{GL}(n)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions $L(1/2,π\otimesπ_0)$, for a fixed cuspidal automorphic representation $π_0$, over the family of $π$ with analytic conductors bounded by a quantity which is tending off to infinity. Our proof uses the integral representations of the $L$-functions, period with regularized Eisenstein series, and the invariance properties of the analytic newvectors.