Jiakun Pan
We prove an asymptotic formula expressing the regularized fourth moment of all newform Eisenstein series of primitive nebentypus as a mean value of $L$-functions. This is an analogue of the recent work of Djanković and Khan in the spectral aspect.
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We obtain an conditional result for the regularized fourth moment of some newform Eisenstein series, using a recipe by Conrey, et al. We also discuss the fourth moment of truncated newform Eisenstein series.
A conjecture for the regularized fourth moment of Eisenstein series
基于Zagier正则化内积，给出了Eisenstein级数四阶矩的随机波动猜想。我们证明了一个渐近公式表示的正则化四阶矩的平均值的L-函数。这是一个优势，在文献中，它已经接近四阶矩问题，通过截断爱森斯坦级数，并没有得到一个合适的表达的L-函数。
Multiple Eisenstein series are holomorphic functions in the complex upper-half plane, which can be seen as a crossbreed between multiple zeta values and classical Eisenstein series. They were originally defined by Gangl-Kaneko-Zagier in 2006, and since then, many variants and regularizations of them have been studied. They give a natural bridge between the world of modular forms and multiple zeta values. In this note, we give a new algebraic interpretation of stuffle regularized multiple Eisenstein series based on the Hopf algebra structure of the harmonic algebra introduced by Hoffman.
We obtain an asymptotic for the regularized fourth moment of the Eisenstein series for the full modular group, in agreement with the Random Wave Conjecture.
Regularized products of Gauss and Eisenstein integers and primes
受Muñoz García和Pérez-Marco工作的启发，采用欧拉式启发式计算，得到了高斯整数和艾森斯坦整数及其素数的正则化无穷乘积的显式表达式。
We consider the Kudla-Millson theta series associated to a quadratic space of signature $(N,N)$. By combining a `see-saw' argument with the Siegel-Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree $N$ is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon-Pozzi-Vonk to totally real fields.
A modular framework for generalized Hurwitz class numbers III
本文扩展了Pei和Wang的广义Cohen-Eisenstein级数到更高权重，并通过正则化theta提升和微分方程与Shintani theta提升相关联。
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…
A note on the sup norm of Eisenstein series
本文证明了实解析Eisenstein级数上的一个上范数有界，其形式为$E(z, 1/2 + iT) \ll T^{3/8 + \varepsilon}$，并且对于$z$在$\mathbb{H}$的一个固定紧子集上是一致的.
$τ$-Norm-Perfect and $τ$-Perfect Eisenstein Integers for $τ=ω+2$ and $2$
利用Robert Spira的复Mersenne数的\cite{D}定义和复因子和函数，刻画了$(ω+2)$ -范数完全数和可被$ω+2$整除的$(ω+2)$ -完满数，证明了在Eisenstein整数中不存在可被$2$整除的$2$ -范数完满数.
In this paper we consider the sup-norm problem in the context of analytic Eisenstein series for $GL(2)$ over number fields. We prove a hybrid bound which is sharper than the corresponding bound for Maaß forms. Our results generalise those of Huang and Xu where the case of Eisenstein series of square-free levels over the base field $\mathbb{Q}$ had been considered.
Shuffle regularization for multiple Eisenstein series of level $N$
本文将 Bachmann 和 Tasaka 关于级数 1 的多重 Eisenstein 级数的洗牌正则化结果推广到任意级数 $N$，并给出级数 $N$ 的 MES 之间的线性关系。
Norm bounds on Eisenstein series
研究特定算术双曲轨形上艾森斯坦级数的范数问题，通过爱泼斯坦zeta函数界和不定二次型受限值计数，为模曲面和皮卡3-流形得出精确指数。
Kronecker limit formulas for parabolic, hyperbolic, and elliptic Eisenstein series via Borcherds products
经典的Kronecker极限公式描述了与模群无穷远尖点相关的非全纯Eisenstein级数在一阶极点处的Laurent展开式中的常数项.最近，Jorgenson，Kramer和第一作者研究了与第一类富克斯群的双曲和椭圆元相关的非全纯Eisenstein级数的亚纯延拓和Kronecker极限型公式.在目前的工作中，我们实现了平均版本的所有三种类型的爱森斯坦系列$Γ_0(N)$作为正则化θ电梯的一个单一类型的庞加莱系列，由于塞尔伯格。利用Poincaré级数的这种实现和性质，我们得到了上述Eisenstein级数的亚纯延拓和Kronecker极限公式。相应的克罗内克极限函数则由与底层庞加莱级数的特殊值相关联的Borcherds乘积的绝对值的对数给出。
In this paper we prove a hybrid subconvexity bound for class group $L$-functions associated to a quadratic extension $K/\mathbb{Q}$ (real or imaginary). Our proof relies on relating the class group $L$-functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the following uniform sup norm bound for Eisenstein series $E(z,1/2+it)\ll_\varepsilon y^{1/2} (|t|+1)^{1/3+\varepsilon}, y\gg 1$, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.
Sup-norm bounds for Eisenstein series
本文在控制谱参数和水平的条件下，给出了上半平面同余式上的Eisenstein级数的界。在这项工作中的关键观察是，我们利用更好的结构的放大器，只是支持素数的爱森斯坦系列，它可以使用分析方法，如杨没有得到一个下界的放大器和几何方法的Harcos-Templier没有获得更有效的治疗计数问题。
We generalise the sum-of-divisors-function $σ$ and evenness to the rings of integers of certain algebraic number fields. In particular, we present necessary and sufficient conditions for even Eisenstein integers to be (norm-)perfect based on the work of McDaniel [McD] on Gaussian integers. Furthermore, some results concerning odd norm-perfect Eisenstein integers and the rings of integers of other cyclotomic fields are proven.
Some norm relations of the Eisenstein classes of GSp(4)
本文在有理数域的分圆扩张中构造了一个Galois上同调类的范数相容系统，从而（间接地）产生了辛群GSp（4）的4次p进L-函数.这些类被定义为椭圆多对数原层的挠截面的上积。我们依赖于规范的兼容性的椭圆polylogram和上同调的西格尔threefolds的一些重量计算。
Primitive Fourier Coefficients of Hermitian Klingen-Eisenstein Series
本文确定了CM扩张上Hermitian Klingen-Eisenstein级数的本原Fourier系数，通过拉回公式将其表示为Siegel Eisenstein级数系数与Petersson内积的乘积，并在假设下证明了$p$-整性。
Pullbacks of Siegel Eisenstein Series and Weighted Averages of Critical $L$-values
本文给出了权$k$和满水平的正规椭圆模形式上的$L$ -函数的特殊值的加权平均公式。这些结果是通过研究SiegelEisenstein级数的拉回并给出显式谱分解得到的。
We obtain pullback formulas for Klingen Eisenstein series with arbitrary levels, with respect to both Siegel congruence and paramodular subgroups, in degree two. Pullback results are used, along with the Fourier series expansion of Klingen Eisenstein series given by Mizumoto, to prove certain identities involving critical values of $ L$-functions attached to normalized elliptic modular forms of weight $ k $ and full level.
The algebra of bi-brackets and regularised multiple Eisenstein series
本文研究了一类称为双括号的$q$ -级数的代数，其系数由分块上的加权和给出。这些级数结合了完整模群的模形式理论以及多重zeta值（MZV）理论，因为它们出现在正则化多重爱森斯坦级数的傅里叶展开中。使用共轭的分区，我们得到双括号之间的线性关系，称为分区关系，自然产生两种不同的方式来表示类似于多个zeta值的填充和洗牌产品的两个双括号的产品。双括号是多因子和的生成函数的推广，称为括号，$[s_1,\dots,s_l]$由作者和U。库恩利用Hoffman关于拟混洗代数的结果，利用双括号的代数结构进一步定义了满足MZV的混洗和填塞积公式的$q$ -级数$[s_1,\dots,s_l]^{sh}$和$[s_1,\dots,s_l]^\ast$.在最近的一项工作中，作者和K。Tasaka定义了正则化的多重Eisenstein级数$G^{sh}$，通过使用一个显式的连接到正式的迭代积分上的余积。这些满足shuffle乘积公式。将同样的概念应用于拟混洗代数上的余积，使我们能够定义满足混洗积的多个Eisenstein级数$G^\ast$。我们证明了$G^{sh}$和$G^\ast$都是由MZV和双括号的乘积的线性组合给出的。通过比较这两个正则化的重Eisenstein级数，我们得到了低权重Eisenstein级数的有限重洗牌关系，推广了以前证明的关系。
We study nearly holomorphic Siegel Eisenstein series of general levels and characters on $\mathbb{H}_{2n}$, the Siegel upper half space of degree $2n$. We prove that the Fourier coefficients of these Eisenstein series (once suitably normalized) lie in the ring of integers of $\mathbb{Q}_p$ for all sufficiently large primes $p$. We also prove that the pullbacks of these Eisenstein series to $\mathbb{H}_n \times \mathbb{H}_n$ are cuspidal under certain assumptions.
Let $E_k$ be the normalized Eisenstein series of weight $k$ on $\mathrm{SL}_{2}(\mathbb{Z})$. Let $φ_k$ be the polynomial that encodes the $j$-invariants of non-elliptic zeros of $E_k$. In 2001, Gekeler observed that the polynomials $φ_k$ seem to be irreducible (and verified this claim for $k\leq 446$). We show that $φ_k$ is irreducible for infinitely many $k$.
Image des opérateurs d'entrelacements normalisés et pôles des séries d'Eisenstein
我们在前面的论文中已经展示了如何使用扭曲内窥镜提升来规范化典型群的交织算子。本文证明了在Eisenstein级数理论中感兴趣的情形下，这种算子的象要么是0，要么是不可约表示。作为结果，我们明确地计算点，在某些假设下，平方可积〔的〕表示的爱森斯坦级数是不全纯的阿基米德的地方：在那里，我们主要假设的无穷小字符是完整的和正规的。
An Eisenstein--Kronecker Modular Symbol and Beilinson--Kato Distributions
本文构造了一个Eisenstein-Kronecker模符号，其特化恢复Borisov-Gunnells符号，并通过归一化将其分布等同于Colmez的Beilinson-Kato分布实现，从而建立了循环Manin关系并生成相关线性关系。
Fourier expansion of light-cone Eisenstein series
本文通过给出双曲 n+1 空间上算术格对应艾森斯坦级数的傅里叶系数显式公式，确定了其极点位置与上确界范数，并据此获得球面上有理点计数的新结果。
Weights of modular forms on $\mathrm{SO}^{+}(2,l)$ and congruences between Eisenstein series and cusp forms of half-integral weight on $\mathrm{SL}_{2}$
令$E$是一个1级的半整数权向量值Eisenstein级数，归一化使得系数都在$\mathbb{Z}$中。我们证明了存在一个一级向量值尖点形式$f$，其权值与$E$相同，系数在$\mathbb{Z}$中，它与$E$模$E$的常数项全等。
On coefficient valuations of Eisenstein polynomials
设p > 2为素数，n > m > 0。设pi_n是zeta_{p^n} - 1在C_{p-1}下的范数，使得Z_（p）[pi_n]| Z_（p）是p^{n-1}次离散赋值环的纯分歧扩张. Q（pi_m）上pi_n的极小多项式是一个艾森斯坦多项式，我们给出了它在pi_m处的系数赋值的下界.函数场模拟，介绍了Carlitz和Hayes，以及研究。
We generalize our previous method on subconvexity problem for $\mathrm{GL}_2 \times \mathrm{GL}_1$ with cuspidal representations to Eisenstein series, and deduce a Burgess-like subconvex bound for Hecke characters, i.e., the bound $|L(1/2,χ)| \ll_{\mathbf{F},ε} \mathbf{C}(χ)^{1/4-(1-2θ)/16+ε}$ for varying Hecke characters $χ$ over a number field $\mathbf{F}$ with analytic conductor $\mathbf{C}(χ)$. As a main tool, we apply the extended theory of regularized integral due to Zagier developed in a previous paper to obtain the relevant triple product formulas of Eisenstein series.
We show an asymptotic formula for the L^2 norm of the Eisenstein series restricted to a segment of a geodesic connecting infinity and an arbitrary real. For generic geodesics of this form, the asymptotic formula shows that the Eisenstein series satisfies restricted QUE. On the other hand, for rational geodesics, the Eisenstein series does not satisfy restricted quantum ergodicity. As an application, we show that the zero set of the Eisenstein series intersects every such geodesic segment, provided the spectral parameter is large. We also make analogous conjectures for the Maass cusp forms. In …
Eisenstein series associated with $Γ_0(2)$
本文定义了规范化Eisenstein级数$\mathcal{P}$、$e$和与$Γ_0(2),$相关的$\mathcal{Q}$，并由三角恒等式导出了它们所满足的三个微分方程.利用这三个公式，我们定义了一个依赖于$Γ_0(2)$上模形式权的微分方程，然后利用正交多项式和高斯超几何级数构造了它的模解.我们还构造了一类与三角数有关的无穷级数。最后，我们从一个涉及三角数的公式中导出了一个组合恒等式。
Epstein zeta-functions, subconvexity, and the purity conjecture
证明了一般k元二次型的Epstein zeta函数的次凸性界。这与GL（k）上Eisenstein级数的上范数界有关，当k >= 2时，其上范数指数为（k-2）/8.特别地，如果$k$是奇数，则该指数不在Z/4中，这表明Sarnak的纯度猜想对Eisenstein级数不成立。
For any even integer $k \ge 4$, let $\E_k$ be the normalized Eisenstein series of weight $k$ for $\SL_2(\Z)$. Also let $\D$ be the closure of the standard fundamental domain of the Poincaré upper half plane modulo $\SL_2(\Z)$. F.~K.~C.~Rankin and H. P. F. Swinnerton-Dyer showed that all zeros of $\E_k$ in $\D$ are of modulus one. In this article, we study the critical points of $\E_k$, that is to say the zeros of the derivative of $\E_k$. We show that they are simple. We count those belonging to $\D$, prove that they are located on the two vertical edges of $\D$ and produce explicit intervals …
Critical points of the classical Eisenstein series of weight two
本文完全确定了权为$2$的规范化Eisenstein级数$E_2(τ)$的临界点。虽然$E_2(τ)$不是模形式，但我们的结果表明，$E_2(τ)$在$Γ_{0}(2)$的每个基本域上至多有一个临界点。我们还给出了一个判别基本整环含有$E_2(τ)$临界点的准则。在$Γ_{0}(2)$作用下的Möbius变换下，所有临界点都可以映射到基本域$F_0$中，并且它们的图像密集地包含在三条平滑曲线上.对这些平滑曲线也作了几何解释。结果表明，这些光滑曲线与复格林函数的平凡临界点的退化曲线相吻合。
Eisenstein series modulo prime powers
本文研究了模素数幂的艾森斯坦级数，通过模形式表示并扩展了相关结果。
Integrality of $\mathfrak{p}$-Adic $L$-Functions at Eisenstein Primes
本文研究平凡新形式的$\mathfrak{p}$进$L$函数在 Eisenstein 同余素点处的整性，在温和分歧假设下证明其属于 Iwasawa 代数，并推出主猜想中的整除关系，权$2$情形亦成立。
We study two closely related problems stemming from the random wave conjecture for Maass forms. The first problem is bounding the $L^4$-norm of a Maass form in the large eigenvalue limit; we complete the work of Spinu to show that the $L^4$-norm of an Eisenstein series $E(z,1/2+it_g)$ restricted to compact sets is bounded by $\sqrt{\log t_g}$. The second problem is quantum unique ergodicity in shrinking sets; we show that by averaging over the centre of hyperbolic balls in $Γ\backslash \mathbb{H}$, quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the …
The $L^2$ restriction of the Eisenstein series to a geodesic segment
本文研究了Eisenstein级数$E(z,1/2+iT)$的$L^2$范数，它仅限于连接无穷远和任意实数的测地线的一段。我们猜想，稍微加厚测地线的这种形式，爱森斯坦级数满足限制QUE。我们证明了一个下界，这符合预测的渐近。我们还证明了一个上界，几乎匹配的下界假设黎曼假设（无条件地，尖锐的上界几乎所有的$T$）。最后，我们证明了有理端点测地线的限制QUE猜想。
Regularized spectral expansion of a Rankin--Selberg period and moments
该研究针对GLₙ上极大退化艾森斯坦级数的平方幅度建立正则化谱分解，应用中证明关联GLₙ×GLₙ Rankin–Selberg中心L值二阶矩与GL₂ L值混合矩的部分互反公式，并在广义拉马努金猜想下得到对应渐近公式。
Eisenstein series, $p$-adic modular functions, and overconvergence, II
本研究延续同题前期工作，用统一方法证明了一类由归一化艾森斯坦级数与$p$进Frobenius算子构造的模函数的显式过收敛速率，拓展至素数2、3并改进了已有结果，还优化了$p\neq5$情形的相关结论。
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.
On the zeros of certain modular functions for the normalizers of congruence subgroups of low levels I
本文研究了亏格为零且水平至多为12的同余子群的正规化子所对应的Hecke型Faber多项式的Eisenstein级数的零点和模函数的位置.在第一部分，我们将考虑规范化子的模函数的一般理论。
A spectral interpretation of the zeros of the constant term of certain Eisenstein series
本文考虑非正规化Eisenstein级数的常数项$ϕ_K(y,s)$附加到$\PSL(2,\cO_K)$上，其中$K$是$\Q$或类数为1的虚二次域.本文的主要目的是证明对于每个$a\ge 1$，Dirichlet级数$ϕ_K(a,s)$的零点可以用自然自伴算子$Δ_a$的特征值进行谱解释.这意味着，除了至多两个实零点，$ϕ_K(a,s)$的所有零点都在临界线上，并且所有零点都是单零点。对于$K=\Q$，这是由于Lagarias和铃木和Ki。
On the zeros of certain modular functions for the normalizers of congruence subgroups of low levels II
本文研究了亏格为零且水平至多为12的同余子群的正规化子所对应的Hecke型Faber多项式的Eisenstein级数的零点和模函数的位置.在第二部分中，我们将通过数值计算观察上述函数零点的位置。
Generating series of intersection volumes of special cycles on unitary Shimura varieties
我们形成了一个生成系列的特殊圈的交集的正则化体积的非紧酉志村簇与一个固定的基变化的周期。我们表明，它是一个希尔伯特模形式，通过确定它与θ积分，我们显示收敛，即使参数位于经典的收敛范围外的韦尔。利用Ichino和Gan-Qiu-Takeda的正则化Siegel-Weil公式，证明了该模形式是与SiegelEisenstein级数相关的n次埃尔米特模形式在U（n，n）上的限制.使用的一个基本事实是一个计算显示的Kudla-MillsonSchwartz函数消失在池田地图。
Let $X$ be a smooth, compact, projective Kähler variety and $D$ be a divisor of a holomorphic form $F$, and assume that $D$ is smooth up to codimension two. Let $ω$ be a Kähler form on $X$ and $K_{X}$ the corresponding heat kernel which is associated to the Laplacian that acts on the space of smooth functions on $X$. Using various integral transforms of $K_{X}$, we will construct a meromorphic function in a complex variable $s$ whose special value at $s=0$ is the log-norm of $F$ with respect to $μ$. In the case when $X$ is the quotient of a symmetric space, then the function we construct is a …
Congruences of Eisenstein series of level $Γ_1(N)$ via Dieudonné theory of formal groups
本文基于Katz的代数几何解释，通过将黎曼-希尔伯特对应重新表述为高度1形式$A$-模的Dieudonné理论，给出了水平$Γ_1(N)$艾森斯坦级数同余的新解释，并推广至高度大于1的形式群。
We show that the central value of class group L-functions of CM fields can be expressed in terms of derivatives of real-analytic Hilbert Eisenstein series at CM points. Then, following an idea of Iwaniec and Kowalski we obtain a conditional explicit lower bound of class numbers of CM fields under a weaker assumption. Some results in the proof lead to an effective nonvanishing result for class group L-functions of general CM fields, generalizing the only known ineffective results.
On the local $L^2$-Bound of the Eisenstein Series
本文利用迹公式谱面展开研究约化群酉 Eisenstein 级数的局部 $L^2$ 范数，证明平均多重对数界，并推出同余商的最优提升与 Sarnak–Xue 计数性质。
A modular framework for generalized Hurwitz class numbers II
本文在 $N=p$ 为奇素数时，将半调和 Maass 形式 $\mathcal{G}$ 实现为 $1$ 的正则化 Siegel theta 提升输出，计算其平方指标 Fourier 系数，并建立了其与 $1$ 的所有二次迹之间的显式恒等式。
The fine spectral expansion of the Rankin-Selberg period
本文通过正则化秩-塞尔伯格周期展开，揭示了秩-塞尔伯格球面变种theta级数的谱结构，并为杰克特-拉里斯追迹公式的细谱展开提供了关键理论支撑。
The purpose of the present paper is to study the number theoretic properties of the special values of the spectral zeta functions of the non-commutative harmonic oscillator (NcHO), especially in relation to modular forms and elliptic curves from the viewpoint of Fuchsian differential equations, and deepen the understanding of the spectrum of the NcHO. We study first the general expression of special values of the spectral zeta function $ζ_Q(s)$ of the NcHO at $s=n$ $(n=2,3,\dots)$ and then the generating and meta-generating functions for Apéry-like numbers defined through the analysis of speci…
Higher Siegel--Weil formula for unitary groups: the non-singular terms
本文在酉 shtuka 模叠上构造特殊闭链，证明归一化 Siegel–Eisenstein 级数非奇异 Fourier 系数的 r 阶中心导数等于虚拟维数 0 特殊闭链的度数，从而给出 Kudla–Rapoport 猜想的函数域类比并涵盖所有高阶导数。
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