Zhining Wei
In this work, we establish a zero density result for the Rankin-Selberg $L$-functions. As an application, we apply it to distinguish the holomorphic Hecke eigenforms for $\operatorname{SL}_2(\mathbb{Z}).$
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On the Rankin-Selberg zeta-function
得到了1/2线上Rankin-Selberg ζ函数的近似函数方程。
p-adic L-functions for Rankin-Selberg convolutions over number fields
本文构造了任意数域上GL（n+1）× GL（n）的Rankin-Selberg卷积的分圆p-adic L函数，并证明了它们满足期望的函数方程.
A note on p-adic Rankin--Selberg L-functions
证明了与非普通模形式相关的某些$p$ -adic Rankin-Selberg $L$ -函数的值的插值公式.
The universality of symmetric power L-functions and their Rankin-Selberg L-functions
建立了自守形式的前四个对称幂L函数及其相应的Rankin-Selberg L函数的普适性定理。这推广了Laurincikas & Matsumoto和Matsumoto的一些结果.
In the 80s, Zagier and Jacquet-Zagier tried to derive the Selberg trace formula by applying the Rankin-Selberg method to the automorphic kernel function. Their derivation was incomplete due to a puzzle of the computation of a residue. We solve this puzzle and complete the derivation. The main input is an extension of the theory of regularized integrals invented by Zagier, which is of independent interest.
On some mean square estimates in the Rankin-Selberg problem
本文对涉及全纯尖点形式系数和的渐近公式的经典Rankin-Selberg问题进行了综述。我们还研究了函数$Δ(x;ξ) (0\leξ\le1)$，Rankin-Selberg问题中的误差项加权的$ξ$ -次幂的对数。证明了$Δ(x;ξ)$的均方估计。
Subconvexity for the Rankin-Selberg L-function in both levels
本文得到了Rankin-Selberg L函数在两个层次上的一个次凸性结果。该结果的新特点是将Duke-Friedlander-Iwaniec的放大方法应用于双Peterson-Kuznetsov迹公式。当迹公式覆盖整个$GL_2$谱时，这个次凸性结果是Rankin-Selberg L函数中Hecke特征形选取的无条件条件.
Real zeros and size of Rankin-Selberg L-functions in the level aspect
本文证明了一类Rankin-Selberg L函数的调和软化二阶矩的渐近公式。其中一个主要的新的输入是一个实质性的改善，这是通过解决一个移位卷积问题的平均谱方法的容许长度的软化。第一个结果是Rankin-Selberg L函数在水平方向上的一个新的次凸性界。此外，还构造了无穷多个至多有8个非平凡实零点的Rankin-Selberg L函数，并得到了这类函数的解析秩的一些新的非平凡估计.
On the Rankin--Selberg problem, II
该研究针对Rankin–Selberg问题，改进了相关界的结果，得到了GL(2)全纯及Maass尖点形式傅里叶系数二阶矩的更小误差项，推进了数论领域该问题的研究进展。
We derive integral representations for the Rankin-Selberg L-functions on GL(3) x GL(1) and GL(3) x GL(2) by a process of unipotent averaging at archimedean places. A key feature of our result is that it allows one to fix the choice of test vector at finite places, irrespective of ramification. This enables a new proof of the functional equation for GL(3) x GL(2) Rankin-Selberg L-functions in many cases.
Beyond endoscopy for the Rankin-Selberg L-function
我们试图通过在迹公式中取极限来理解L-函数的极点。该技术避免了内窥镜和Kim-Shahidi方法。特别地，我们研究了Rankin-Selberg L函数的极点。利用解析数论的技巧来取这个极限，我们基本上得到了一个新的证明，证明了Rankin-Selberg L函数在$s=1.$上的解析性。
A standard zero free region for Rankin-Selberg L-functions
得到了Rankin Selberg L函数$L(s, f\times \widetilde{f})$的标准无零区域，其中$f$是$GL(n)$上的几乎处处调和Maass型，且$f$不一定是自对偶的.该方法是基于理论的爱森斯坦系列推广的工作Sarnak。
In this paper, we solve the Rankin--Selberg problem. That is, we break the well known Rankin--Selberg's bound on the error term of the second moment of Fourier coefficients of a $\mathrm{GL}(2)$ cusp form (both holomorphic and Maass), which remains its record since its birth for more than 80 years. We extend our method to deal with averages of coefficients of L-functions which can be factorized as a product of a degree one and a degree three L-functions.
We present a Rankin-Selberg integral on the exceptional group $G_2$ which represents the L-function for generic cuspidal representations of $\widetilde{SL}_2\times GL_2$. As an application, we show that certain Fourier-Jacobi type periods on $G_2$ are non-vanishing.
Landau-Siegel zeros of Rankin-Selberg $L$-functions
本文研究了兰道-塞尔伯格 $L$-函数和三重积 $L$-函数在无异常兰道-西格尔零点的零自由区域中的性质。
We prove a version of the Extra-zero conjecture formulated by the first named author for p-adic L-functions associated to Rankin-Selberg convolutions of modular forms of the same weight. The novelty of this result is to provide strong evidence in support of this conjecture in the non-critical case, which remained essentially unstudied.
On effective determination of Maass forms from central values of Rankin-Selberg $L$-function
本文讨论了从Rankin-Selberg $L$ -函数$L(1/2,f\otimes h)$的中心值确定满能级的Hecke-Maass尖点形式$f$的问题，其中$h$贯穿满能级的Hecke-Maass本征形集.我们在这个方向上证明了一个定量结果。
Special values of anticyclcotomic Rankin-Selberg L-functions
本文证明了复曲面Hilbert模形式的Waldspurger公式。作为应用，我们构造了Hilbert模形式的一类反分圆p-adic Rankin-Selberg L函数，推广了Bertolini，Darmon和Prasanna在椭圆情形下的构造.此外，在Hida的工作基础上，我们给出了该p-adic L-函数的Iwasawa μ-不变量为零的充要条件，并证明了中心Rankin-Selberg L-值的反分圆扭模$p$的非零性.
Bessel periods and Selmer groups over ordinary Rankin--Selberg eigenvariety
本文引入酉群及其Rankin--Selberg乘积的普通分布概念以重新定义普通特征簇，在定与不定情形下构造Bessel周期，并提出关联其消失除子与Selmer群特征除子的Iwasawa型猜想，证明了部分整除关系。
Period integrals and Rankin-Selberg L-functions on GL(n)
本文计算了一类Rankin-Selberg $L$ -函数L（f x g，1/2）的二阶矩，其中f和g是GL（n）上的Hecke-Maass尖点形式.我们的边界是一样强的Lindelöf假设的平均，并恢复个别的凸性边界。这一结果即使在经典的n=2的情况下也是新的。
On the Rankin-Selberg problem in families
本文在全纯模形式和Hecke-Maass尖点形式族的短区间上研究Rankin-Selberg问题，对前者假设平均Lindelöf界，对后者不作假设。
We prove new congruences between special values of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and to prove the existence of non-abelian $p$-adic $L$-functions for Hida families on $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$. As an application, we prove strong non-vanishing results for central $L$-values: We give sufficient local conditions for twisted central Rankin-Selberg $L$-values to be generically non-zero.
Congruences for the ratios of Rankin--Selberg $L$-functions
本文通过计算研究Rankin-Selberg L-函数比值的同余关系，提出一个普遍的精确猜想。
We give a simple proof of a standard zero-free region in the $t$-aspect for the Rankin--Selberg $L$-function $L(s,π\times \widetildeπ)$ for any unitary cuspidal automorphic representation $π$ of $\mathrm{GL}_n(\mathbb{A}_F)$ that is tempered at every nonarchimedean place outside a set of Dirichlet density zero.
The Second Moment of Rankin-Selberg $L$-Functions in Conductor-Dropping Regimes
研究Rankin-Selberg L-函数在降阶 regime 中二次矩的渐进行为，采用holomorphic Hecke cusp forms等方法，得到渐近公式。
We prove a functional equation for the three-variable $p$-adic $L$-function attached to the Rankin-Selberg convolution of a Coleman family and a CM Hida family, which was studied by Loeffler and Zerbes. Consequentially, we deduce that an anticyclotomic $p$-adic $L$-function attached to a $p$-non-ordinary modular form vanishes identically in the indefinite setting.
On the Rankin-Selberg problem: higher power moments of the Riesz mean error term
令$Δ_1(x;ϕ)$为Rankin-Selberg zeta函数的第一Riesz均值的误差项。研究了$Δ_1(x;ϕ)$的高阶矩，并利用Ivić的大值论证导出了三阶、四阶和五阶矩的渐近公式.
In this paper, we study the non-vanishing of the central values of the Rankin-Selberg $L$-function of two adèlic Hilbert primitive forms ${\bf f}$ and ${\bf g}$, both of which have varying weight parameter $k$. We prove that, for sufficiently large $k$, there are at least $\frac{k}{(\log k)^{c}}$ adèlic Hilbert primitive forms ${\bf f}$ of weight $k$ for which $L(\frac12, {\bf f}\otimes{\bf g})$ are nonzero.
Modular symbols for reductive groups and p-adic Rankin-Selberg convolutions over number fields
我们给出了一个广泛的一类模块化符号的结构连接到约化群。作为应用，我们构造了一个p-adic分布，它内插数域上GL（n）和GL（n-1）的尖形自守表示上的扭曲Rankin-Selberg L-函数的特殊值.如果表示在p是普通的，我们的分布是有界的，并产生相关的p-adic L-函数的解析性。
The central value of the Rankin-Selberg $L$-functions
设$f$是$SL(3, \mathbb{Z})$的固定的Maass型，$u_j$是$SL(2, \mathbb{Z}),$的偶Maass型的规范正交基，我们证明了$f$和$u_j$的Rankin-Selberg $L$ -函数与$u_j$的$L$ -函数的乘积在中心值1/2处的平均值的渐近公式.这意味着这些$L$ -函数在$1/2.$的同时非零结果
We prove a new case of spectral reciprocity formulae for the product of $GL(n+1) \times GL(n)$ and $GL(n) \times GL(n-1)$ Rankin-Selberg $L$-functions ($n \geq 3$), which are first developed by Blomer and Khan in \cite{BK17} for degree 8 $L$-functions ($n=2$ case, the product of $GL(3) \times GL(2)$ and $GL(2) \times GL(1)$ Rankin-Selberg $L$-functions). Our result can be viewed as a generalization of Blomer and Khan's work to higher rank case. We will mainly follow the method developed in \cite{Nun20}. We will use the integral representations of Rankin-Selberg $L$-functions generalized by Ich…
On the fourth moment in the Rankin-Selberg problem
若经典Rankin-Selberg问题的误差项为$$ Δ(x) := \sum_{n\le x}c_n - Cx $$，则证明了$$ \int_0^X Δ^4(x)\d x \ll_εX^{3+ε},\quad \int_0^X Δ_1^4(x)\d x \ll_εX^{11/2+ε}, $$，其中$Δ_1(x) = \int_0^xΔ(u) du$ .后一个界限最大为“$ε$”，是可能的最佳界限。
Weight aspect asymptotic and simultaneous non-vanishing for Rankin-Selberg $L$-functions
本文通过在权重方面证明渐进行为的结果，展示两个Rankin-Selberg L-函数的同时非消失性，并改进误差项的平方根指数。
Triple product integrals and Rankin-Selberg L-functions
本文证明了一个互反公式，将权0和1/2自守形式的三重积积分谱平均与权0自守形式的经典Rankin-Selberg积分相关联。
Determining Hilbert Modular Forms by Central Values of Rankin-Selberg Convolutions: The Level Aspect
本文证明了一个本原Hilbert尖点形式$\mathbf{g}$是由Rankin-Selberg $L$ -函数$L(\mathbf{f}\otimes\mathbf{g}, \frac{1}{2})$的中心值唯一确定的，其中$\mathbf{f}$对无穷多个素理想$\mathfrak{q}$贯穿所有水平$\mathfrak{q}$的本原Hilbert尖点形式.这一结果是将Luo的一个定理推广到全实数域的情形。
On the higher moments of the error term in the Rankin-Selberg problem
本文研究了经典Rankin-Selberg问题中误差项的高阶幂矩，导出了其3次、4次和5次幂矩的渐近公式，改进了已有结果。
We formulate and for the most part prove a conjecture in the style of Mazur-Greenberg for the nonvanishing of central values of Rankin-Selberg $L$-functions attached to elliptic curves in abelian extensions of imaginary quadratic fields. This in particular generalizes the theorem of Rohrlich on $L$-functions of elliptic curves in cyclotomic towers to the setting of abelian extensions of imaginary quadratic fields, corresponding to families of degree-four $L$-functions given by $\operatorname{GL}(2)\times\operatorname{GL}(2)$ Rankin-Selberg $L$-functions. It also generalizes the theorems of Roh…
The Ranking-Selberg integral on ${\bf GSp(2)}$ for square free levels
本文计算了Piatetski-Shapiro在adeles上引入的Rankin-Selberg类型积分，针对平方自由水平的Siegel cuspidal形式向量值，同时给出了特定测试函数在Bessel模型中不可约可容表示的局部zeta积分精确求值。
The Weyl bound for Rankin-Selberg $L$-functions with Joint Ramification
本文通过改进的追踪公式和p进 stationary 相位方法，建立了Rankin-Selberg L-函数在联合分歧设定下的Weyl界，并获得了Whittaker函数乘积积分的精确界。
Eisenstein cohomology and congruences for the ratios of Rankin--Selberg $L$-functions
本文利用改进的埃森斯坦上同调理论，证明了Rankin-Selberg L函数临界值比值与对象之间的同余关系。
Euler systems for Rankin--Selberg convolutions of modular forms
利用模曲线乘积的高阶Chow群中的元素，构造了两个模形式的伽罗瓦表示的张量积的上同调的Euler系统.当相关的p-adic Rankin-Selberg L-函数在s = 1处不为零时，我们使用这个欧拉系统证明了伽罗瓦表示的严格Selmer群的有限性定理。
The conjectural newform theory for generic representations of $p$-adic ${\rm SO}_{2n+1}$ was formulated by P.-Y. Tsai in her thesis in which Tsai also verified the conjecture when the representations are supercuspidal. The main purpose of this work is to compute the Rankin-Selberg integrals for ${\rm SO}_{2n+1}\times{\rm GL}_r$ with $1\le r\le n$ attached to newforms and also oldforms under the validity of the conjecture.
Stable averages of central values of Rankin-Selberg L-functions: some new variants
如Michel-Ramakrishan（2007）和Feigon-Whitehouse（2008）所证明的，对于固定偶数权和大水平的一些全纯形式的Rankin-Selberg卷积的平均中心L值，存在固定虚二次θ级数的“稳定”公式。我们得到确切的有限公式扭曲的第一时刻的Rankin-Selberg L-值在更大的一般性，并证明类似的“稳定”公式，当一个考虑任意模块扭曲的大素数的权力水平或真正的二面角扭曲的奇数类型的混合签名的Hecke字符。
Non-vanishing modulo p of central critical Rankin-Selberg L-values with anticyclotomic twists
证明了$\ell$ -功率导体的中心临界Rankin-Selberg L-值的模p，对于一个不同于p的素数$\ell$，具有反分圆扭转的非零模p. L-函数是虚二次域的尖点形式与算术Hecke特征标的θ级数的Rankin乘积。本文讨论了Hecke特征标的权大于尖点形式的权时的L-值与Vatsal和Cornut-Vatsal在同一主题下的标志性著作中的L-值有本质的不同。
Let $F$ be a number field, and let $π_1$ and $π_2$ be distinct unitary cuspidal automorphic representations of $\operatorname{GL}_{n_1}(\mathbb{A}_F)$ and $\operatorname{GL}_{n_2}(\mathbb{A}_F)$ respectively. In this paper, we derive new lower bounds for the Rankin-Selberg $L$-function $L(s, π_1 \times \widetildeπ_2)$ along the edge $\Re s = 1$ of the critical strip in the $t$-aspect. The corresponding zero-free region for $L(s, π_1 \times \widetildeπ_2)$ is also determined.
Let $F$ be a non-Archimedean locally compact field. Let $σ$ and $τ$ be finite-dimensional semisimple representations of the Weil-Deligne group of $F$. We give strong upper and lower bounds for the Artin and Swan exponents of $σ\otimesτ$ in terms of those of $σ$ and $τ$. We give a different lower bound in terms of $σ\otimes\checkσ$ and $τ\otimes\checkτ$. Using the Langlands correspondence, we obtain the bounds for Rankin-Selberg exponents.
A hybrid asymptotic formula for the second moment of Rankin-Selberg L-functions
本文研究了满水平固定模形式的Rankin-Selberg L函数和大偶数权、固定水平、固定本原Neumanypus的全纯尖点新形式。我们在相当一般的范围内计算了这个族的二阶矩，并得到了一个具有省电误差项的渐近公式。一个特殊的情况处理的四阶矩的L函数与全纯尖点形式。
In this work we use the Rankin-Selberg method to obtain results on the analytic properties of the standard $L$-function attached to vector valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted $L$-function beyond the usual range of absolute convergence. We remark that these results were known in this generality only in the case of scalar weight Siegel modular forms. As an interesting by-product of our work we establish the cuspidality of some theta series.
Zeroes of Rankin-Selberg $L$-functions and the trace formula
本文用Rankin-Selberg L-函数的零点表示GL(n)迹公式谱侧中极大抛物爱森斯坦级数的贡献，推广了GL(2)情形，并由此得到零点之和的下界及n=2时素数阶循环扩张的基变换关系。
Rankin--Selberg coefficients in arithmetic progressions modulo prime powers
研究模素数幂的Rankin-Selberg系数分布，基于Ramanujan-Petersson猜想，证明系数具有特定分布水平。
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.
Determining Hilbert Modular Forms by Central Values of Rankin-Selberg Convolutions: The Weight Aspect
本文证明了一个本原Hilbert尖点形式$\mathbf{g}$是由Rankin-Selberg $L$ -函数$L(\mathbf{f}\otimes\mathbf{g}, \frac{1}{2})$的中心值唯一确定的，其中对于无穷多个权向量$k$，$\mathbf{f}$贯穿所有权为$k$的本原Hilbert尖点形式.这项工作是一个推广的结果Ganguly，Hoffstein和Sengupta的设置全实数域，它是一个重量方面的模拟作者最近的工作。
Uniqueness of Rankin-Selberg products
本文证明了由Jacquet，Piatetski-Shapiro和Shalika定义的$γ$ -因子与由Langlands-Shahidi方法得到的因子相等.我们的结果是新的情况下，积极的特点，我们建立了一个改进的版本的局部-整体原则的${\rm GL}_n$，有独立的利益。在特征零中，结果是由于沙希迪。通过Rankin-Selberg $γ$-因子的唯一性结果，对$γ$-因子进行了比较.
Level Aspect Subconvexity For Rankin-Selberg $L$-functions
设$M$是一个无平方数的整数，且$P$是一个不整除$M$的素数，使得$P \sim M^η$与$0<η<2/21$。当$f$和$g$分别是水平$P$和$M$的两个本原全纯尖点形式时，证明了$L(\tfrac{1}{2}, f \otimes g)$的次凸性界.这些界限是通过未放大的二阶矩方法实现的。
On the Rankin-Selberg problem in short intervals
如果$$ Δ(x) \;:=\; \sum_{n\leqslant x}c_n - Cx\qquad(C>0) $$表示经典Rankin-Selberg问题中的误差项，那么对于一定范围的$U = U(X)$，我们得到了$Δ(x+U) - Δ(x)$的均方的一个非平凡上界.特别是，根据Lindelöf假设的$ζ(s)$，它表明，$$ \int_X^{2X} \Bigl(Δ(x+U)-Δ(x)\Bigr)^2\,{\roman d} x \;\ll_ε\; X^{9/7+ε}U^{8/7}, $$，而根据Lindelöf假设的Rankin-Selberg zeta函数的积分是有界的$X^{1+ε}U^{4/3}$。对于$Δ(x+U)-Δ(x)$的离散二阶矩的类似结果也成立。
Let $f$ be a Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplace eigenvalue $1/4+μ_f^2$, $μ_f>0$. Let $g$ be an arbitrary but fixed holomorphic or Maass cusp form for $\rm SL_2(\mathbb{Z})$. In this paper, we establish the following uniform subconvexity bound for the Rankin-Selberg $L$-function $L(s,f\otimes g)$ $$ L\left(1/2+it,f\otimes g\right)\ll (μ_f+|t|)^{9/10+\varepsilon}, $$ where the implied constant depends only on $\varepsilon$ and $g$.
In this article we obtain an explicit formula for certain Rankin-Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the composition of the Ikeda lift and the Eichler-Zagier-Ibukiyama correspondence. The integral weight version of the main theorem had been obtained by Katsurada and Kawamura. The result of the integral weight case is a product of $L$-function and Riemann zeta functions, while half-integral weight case is a infinite summation over negative fundamental discriminants w…
Rationality results are proved for the ratios of critical values of Rankin-Selberg L-functions of GL(n) x GL(n') over a totally imaginary field F, by studying rank-one Eisenstein cohomology for the group GL(N)/F, where N = n+n', generalizing the methods and results of previous work with Guenter Harder where the base field was totally real. In contrast to the totally real situation, the internal structure of the totally imaginary base field has a delicate effect on the rationality results.
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.
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…
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