The $L^2$ restriction norm of a $GL_3$ Maass form
Xiaoqing Li, Matthew P. Young
我们证明了一个尖锐的上界的$GL_3$马斯形式限制到$GL_2 \times \mathbb{R}^{+}$的$L^2$范数。
We prove a sharp upper bound on the $L^2$ norm of a $GL_3$ Maass form restricted to $GL_2 \times \mathbb{R}^{+}$.
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The $L^2$ restriction norm of a Maass form on $SL_{n+1}(\mathbb{Z})$
讨论了$\SL{n+1}{Z}$上的Maass型的$L^2$限制范数的上、下界.
Restrictions of SL_3 Maass forms to maximal flat subspaces
设n是\Q上三次除代数上的Hecke-Maass型.我们应用算术放大来改进限制在极大平坦子空间的L^2范数的局部界。
Nodal domains of Maass forms I
本文讨论了Hejhal和Rackner在1992年关于Maass形式的数值计算中提出的一些问题。我们建立尖锐的上限和下界的$L^2$ -限制这些形式的某些曲线的模块化表面。这些结果，连同Lindelof假设和已知的次凸$L^\infty$ -边界被应用于证明，当地的节点域的数量这样的形式与其特征值趋于无穷大。
Nodal domains of Maass forms II
在第一部分中，我们给出了一个多项式增长下界的节点域的数目的Hecke-Maass尖形在一个紧凑的一部分，模曲面，假设Lindelöf假设。这是拓扑论证和已知的次凸性估计以及马斯形式的新的尖锐下界限制定理的结果。本文讨论了同样的问题，一般（紧凑或不）算术曲面具有反射对称。拓扑参数的扩展和表示理论的方法需要的限制定理，以及Waldspurger的结果。各种明确的例子给出和研究。
This work contains a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a SL(3,Z) Hecke-Maass cusp form restricted to a compact set.
On the sup-norm of Maass cusp forms of large level. III
设$f$是无平方能级$N$和拉普拉斯特征值$λ$的Hecke-Maass尖点新形式.证明了对于任何$ε>0$，$\pnorm{f}_\infty \ll_{λ,ε} N^{-1/6}+ε} \pnorm{f}_2$.
We prove an upper bound for geodesic periods of Maass forms over hyperbolic manifolds. By definition, such periods are integrals of Maass forms restricted to a special geodesic cycle of the ambient manifold, against a Maass form on the cycle. Under certain restrictions, the bound will be uniform.
Local bounds for $L^p$ norms of Maass forms in the level aspect
本文应用调和分析的技巧研究了四元数除代数上变阶Maass型的$L^p$范数。我们的第一个结果给出了一个候选人的本地界的超级规范的水平，这是新的水平时，不是squarefree。第二个结果是$L^p$规范的水平方面，类似于Sogge的定理$L^p$规范的拉普拉斯特征函数。
Summation Formulas for Harmonic Maass Forms
本文利用调和Maass形式的L-级数理论，去掉了增长限制，证明了更一般的求和公式，并应用于分拆函数及经典模形式的Riesz和，得到求和公式与渐近性质。
In this paper, we study non-trivial upper bounds for the sum $\sum \limits_{n \in S} |λ_f(n)|$ where $f$ is a normalized Maass eigencusp form for the full modular group, $λ_f(n)$ is the $n$th normalized Fourier coefficient of $f$ and $S$ is a proper subset of positive integers in $[1,x]$ with certain properties.
Kakeya-Nikodym norms of Maass forms on $\rm{U}(2,1)$
本文通过放大方法，在复双曲面上研究Maass形式的Kakeya-Nikodym范数，并改进了Sogge的局部界。
Quantitative quantum ergodicity and the nodal domains of Maass-Hecke cusp forms
本文证明了模曲面上Hecke-Maass尖点形式的量子遍历性的一个定量表述。作为结果的一个应用，沿着偶Hecke-Maass尖点形式的密度$1$子序列，我们得到了$η=\{iy~:~y>0\} \subset \mathbb{H}$的固定紧测地线段的约束的$L^2$ -范数的一个尖锐的下界.我们还获得了一个上界的$O_ε\left(t_ϕ^{3/8+ε}\right)$的$L^\infty$规范沿密度$1$子序列的Hecke-马斯尖点形式，对于这样的形式，这是一个改进的Iwaniec和Sarnak给出的上界的$O_ε\left(t_ϕ^{5/12+ε}\right)$。在Ghosh、Reznikov和Sarnak最近的工作中，作者证明了对于所有偶数Hecke-Maass形式，与$η$的测地线段相交的节点域的数量比任何$ε>0$的$t_ϕ^{1/12-ε}$增长得更快，假设Lindel{<$}f假设是真的，并且测地线段足够长。在删除密度为零的子集，甚至Hecke-马斯形式，我们证明没有任何假设，节点域的数量增长速度比$t_ϕ^{1/8-ε}$任何$ε>0$。
We prove a power saving over the local bound for the sup norm of Hecke-Maass forms on any quasi-split semisimple real group that is not isogenous to a product of odd special unitary groups.
Restrictions of Maass forms on $\mathrm{SL}(2,\mathbb{C})$ to hyperbolic surfaces and geodesic tubes
本文通过算术放大方法改进了Maass形式在超曲面和测地管上的L²界，获得了更优的功率节省结果。
In this paper, we improve the sup-norm bound and the lower bound of the number of nodal domains for dihedral Maass forms, which are a distinguished sequence of Laplacian eigenfunctions on an arithmetic hyperbolic surface. More specifically, let $ϕ$ be a dihedral Maass form with spectral parameter $t_ϕ$, then we prove that $\|ϕ\|_\infty \ll t_ϕ^{3/8+\varepsilon} \|ϕ\|_2$, which is an improvement over the bound $t_ϕ^{5/12+\varepsilon} \|ϕ\|_2$ given by Iwaniec and Sarnak. As a consequence, we get a better lower bound for the number of nodal domains intersecting a fixed geodesic segment under the…
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.
In this work we prove a prime number type theorem involving the normalised Fourier coefficients of holomorphic and Maass cusp forms, using the classical circle method. A key point is in a recent paper of Fouvry and Ganguly, based on Hoffstein-Ramakrishnan's result about the non-existence of the Siegel zeros for $GL(2)$ $L$-functions, which allows us to improve preceding estimates.
We investigate the norm of a degree 2 Siegel modular form of asymptotically large weight whose argument is restricted to the 3-dimensional subspace of its imaginary part. On average over Saito-Kurokawa lifts an asymptotic formula is established that is consistent with the mass equidistribution conjecture on the Siegel upper half space as well as the Lindelof hypothesis for the corresponding Koecher-Maass series. The ingredients include a new relative trace formula for pairs of Heegner periods.
Hybrid sup-norm bounds for Maass newforms of powerful level
设$f$是水平$N$、特征标$χ$和Laplace特征值$λ$的$L^2$ -归一化Hecke-Maass尖点新形式.令$N_1$表示使得$N|N_1^2$的最小整数，并且$N_0$表示使得$N_0^2 |N$的最大整数。令$M$表示$χ$的导体并定义$M_1= M/\gcd(M,N_1)$。在本文中，我们证明了有界$|f|_\infty$ $\ll_ε$ $N_0^{1/6 + ε} N_1^{1/3+ε} M_1^{1/2} λ^{5/24+ε}$，它推广和加强以前已知的上界$|f|_\infty$。这是第一次混合边界（即，涉及$N$和$λ$两者）的情况下，已经为$|f|_\infty$建立了非无平方$N$的情况。在非无平方数的情况下，唯一已知的界限是在N方面;作者已经证明，$|f|_\infty \ll_{λ, ε} N^{5/12+ε}$提供了$M=1$。在上述情况下，本结果显着改善了$N$的指数。如果$N$是一个无平方的整数，我们的界限减少到$|f|_\infty \ll_εN^{1/3 + ε}λ^{5/24 + ε}$，这是以前证明的Templier。本工作的关键新特点是系统地使用p-adic表示理论技术，特别是详细研究了Whittaker newforms和矩阵系数为$GL_2(F)$ $F$是一个本地字段。
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 …
Duke, Imamoglu, and Toth constructed a polyharmonic Maass form of level 4 whose Fourier coefficients encode real quadratic class numbers. A more general construction of such forms was subsequently given by Bruinier, Funke, and Imamoglu. Here we give a direct construction of such a form for the full modular group and study the properties of its coefficients. We give interpretations of the coefficients of the holomorphic parts of each of these polyharmonic Maass forms as inner products of certain weakly holomorphic modular forms and harmonic Maass forms. The coefficients of square index are part…
Beyond the spherical sup-norm problem
本文针对极大紧子群非交换时的非球面 Maass 形式提出并求解了 SL_2(C) 算术商上的 sup 范数问题，达到亚 Wigner 指数，并发展了广义球面函数的解析理论与矩阵计数问题。
Geodesic restrictions of arithmetic eigenfunctions
设X是算术双曲曲面，L是X上的测地线段，L是Hecke-Maass形式。通过扩展Iwaniec和Sarnak提出的算术放大技术，我们得到了在Burq-Gérard-Tzvetkov的局部界上对于限制为l的L^2范数的功率节省。我们还改进了沿l的各种傅里叶系数的局部界。
Let $f$ be a Petersson normalized Hecke-Maass cusp form with spectral parameter $t\geq 2$ and let $\mathcal{C}_{D}$ be the union of closed geodesics in $\text{Sl}_{2}(\mathbb{Z})\setminus \mathbb{H}$ associated to a fundamental discriminant $D>0$. Following a suggestion by Sarnak in his letter to Reznikov, we express the restriction norm $||f|_{\mathcal{C}_{D}}||_{2}^{2}$ as a weighted sum of central values of L-functions using Waldspurger's formula. This allows us to get an unconditional improvement over the current bounds.
A Restriction Norm Problem for Siegel Modular Forms
本文建立了2次Siegel尖形式在虚轴上平均$L^2$-范数的渐近公式，误差项有幂次节省，与质量等分布猜想和Lindelöf假设一致，并细致分析了Kitaoka公式。
Low-lying zeroes of Maass form $L$-functions
Katz-Sarnak密度猜想指出，自守$L$ -函数族零点分布的标度极限与酉群$U(N)$的经典子群特征值分布的标度极限一致。这个猜想经常通过计算特定的统计量来测试，例如一级密度，它在中心点附近的归一化零点处用紧支持的傅里叶变换来评估测试函数。Iwaniec、Luo和Sarnak研究了权$k$和水平$N$的尖瓣新形式的一级密度。在极限为$kN \to\infty$的情形下，证明了在$(-2,2)$支持的傅里叶变换下，这些族具有与正交型相一致的单能级密度.超过$(-1,1)$是重要的，因为三个正交基团对于高达$(-1,1)$的支持是不可区分的，但是对于任何更大的支持是可区分的。我们研究了$\mathbb{Q}$上的另一类${\rm GL}_2$自守形式：Maass形式.为了便于分析，我们在库兹涅佐夫公式中使用光滑的权重函数，除了其他限制之外，该函数在原点处消失以订购PDMATHAMBMV。对于在$\left(-2 + \frac{2}{2M+1}, 2 - \frac{2}{2M+1}\right)$中支持Fourier变换的测试函数，我们无条件地证明了当特征值趋于无穷大时，第一级Maass形式的低零点的一级密度仅与正交矩阵的标度极限的一级密度一致.
Let $f$ and $g$ be two Hecke-Maass cusp forms of weight zero for $SL_2(\mathbb Z)$ with Laplacian eigenvalues $\frac{1}{4}+u^2$ and $\frac{1}{4}+v^2$, respectively. Then both have real Fourier coefficients say, $λ_f(n)$ and $λ_g(n)$, and we may normalize $f$ and $g$ so that $λ_f(1)=1=λ_g(1)$. In this article, we first prove that the sequence $\{λ_f(n)λ_g(n)\}_{n \in \mathbb{N}}$ has infinitely many sign changes. Then we derive a bound for the first negative coefficient for the same sequence in terms of the Laplacian eigenvalues of $f$ and $g$.
On the trace of Hecke Operators for Maass forms for congruence subgroups
设E_lambda是一个希尔伯特空间，其元素是Laplace-Beltrami算子的本征函数所张成的函数，本征值λ>0.这个空间中元素的范数由Petersson内积给出。本文计算了空间E_lambda上的Hecke算子T_n对无平方水平的Gamma_0（N）的同余子群的迹，它可以看作是权为零的非全纯尖点形式的Eichler-Selberg迹公式[11]的类似。
H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Some Indefinite Theta Series
在以前的工作中，Zwegers（和Zagier）定义的一元$\widehatμ$ -函数和他的签名格$(r\!+\!1,1)$上的不定θ级数都是海森堡调和的Maaesthet-Jacobi形式。本文将Heisenberg调和性的概念推广到任意多个椭圆变量的Maaubrill-Jacobi形式，得到了符号为$(r\!+\!s,s)$的非退化格的“乘积型”的不定θ级数.因此，我们得到一个干净的推广$\widehatμ$这些负定格。通过对Heisenberg调和Maaesthet-Jacobi型扭点的限制，得到了在Zagier和Zwegers意义下的高阶调和弱Maaesthet-Jacobi型。特别是，我们解释了一些模块化完成，所谓的退化不定θ系列的背景下，更高的深度混合模拟模形式。本文发展的Heisenberg调和Maaesthet-Jacobi型的结构理论也解释了Zwegers二元$\widehatμ$ -函数奇异分裂为亚纯Jacobi型和一元Maaesthet-Jacobi型之和的现象。
Let $f$ be a Hecke-Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplace eigenvalue $λ_f(Δ)=1/4+μ^2$ and let $λ_f(n)$ be its $n$-th normalized Fourier coefficient. It is proved that, uniformly in $α, β\in \mathbb{R}$, $$ \sum_{n \leq X}λ_f(n)e\left(αn^2+βn\right) \ll X^{7/8+\varepsilon}λ_f(Δ)^{1/2+\varepsilon}, $$ where the implied constant depends only on $\varepsilon$. We also consider the summation function of $λ_f(n)$ and under the Ramanujan conjecture we are able to prove $$ \sum_{n \leq X}λ_f(n)\ll X^{1/3+\varepsilon}λ_f(Δ)^{4/9+\varepsilon} $$ with the implied constant depending only …
$L^4$-norms and sign changes of Maass forms
本文无条件证明了Maass形式的$L^4$-norms的Iwaniec-Sarnak猜想，并研究了其符号变化性质。
Let $G$ be an anisotropic semisimple group over a totally real number field $F$. Suppose that $G$ is compact at all but one infinite place $v_0$. In addition, suppose that $G_{v_0}$ is $\mathbb{R}$-almost simple, not split, and has a Cartan involution defined over $F$. If $Y$ is a congruence arithmetic manifold of non-positive curvature associated to $G$, we prove that there exists a sequence of Laplace eigenfunctions on $Y$ whose sup norms grow like a power of the eigenvalue.
Petersson norms of Borcherds theta lifts to O(1, 8n+1) with applications to injectivity and sup-norm bounds
本文研究一级 Maass 尖点形式经 Borcherds theta 提升至 O(1,8n+1) 后的 Petersson 范数，给出含阿基米德局部因子的显式公式，并据此证明提升的单射性及依 Laplace 特征值的 sup 范数界。
On the 4-norm of an automorphic form
我们证明了最优上界和||F||_4^4 << q^{eps}其中f在素数水平q和有界谱参数的Maass尖点形式的规范正交基上运行。
Subconvexity and equidistribution of Heegner points in the level aspect
设q是素数，-D < -4是奇基本判别式，使得q在Q（\sqrt{-D}）中分裂。对于f是q层的权为零的Hecke-Maass新形式，h是D层的权为1 θ的级数，对应于Q（\sqrt{-D}）的一个理想类群特征标，当q = D^η且0 < η< 1时，我们建立了L（f \times h，s）在中心点的混合次凸界.通过这个循环，我们证明了当q，D变大且q < D^{1/20- \varepsilon }时，水平q和判别式D的Heegner点在自然意义上变得等分布。我们解决这些问题的方法与估计限制于Heegner点集的大水平Maass形式的L^2-限制范数有关。进一步建立了同时具有大水平和大二次扭曲的Hecke-Maass L-函数的二次扭曲的界，以及二次Dirichlet L-函数在一定范围内的混合界.
The sup-norm problem for the Siegel modular space of rank two
设F是Siegel模群Sp（4，Z）的秩为2的Siegel上半空间上的平方可积〔的〕Maass型，其Laplace特征值为λ.如果，此外，F是一个联合特征函数的Hecke代数，我们显示了一个节能的超范数界的λ（相对于一般的界）的F限制到一个紧集。作为一个独立的辅助结果，我们证明了半单李群上球函数的新的一致界。
Weighted geodesic restrictions of arithmetic eigenfunctions
本文通过算术放大方法，改进了算术双曲面上特征函数的加权测地限制界，并在一般二维黎曼流形上获得了特征函数的凯凯亚-尼科迪姆界。
The sup-norm problem for PGL(4)
设F是SL（4，Z）的Hecke-Maass尖点形式，其Laplace特征值为λ.假设F在无穷远处满足拉马努金猜想（几乎所有的尖点形式都满足）。我们示出了一个节能的超范数界的lambda（相对于一般的界）F限制到一个紧凑的集合。
Bounding sup-norms of cusp forms of large level
设f是无平方数水平N的$L^2$归一化权零Hecke-Maass尖点形式，特征$χ$和Laplacian特征值$λ\geq 1/4$ .结果表明，$\| f \|_{\infty} \ll_λ N^{-1/37}$，从其中的混合绑定$\|f \|_{\infty} \ll λ^{1/4} (Nλ)^{-δ}$（对于某些$δ> 0$）的衍生。第一个界限也适用于$f = y^{k/2}F$，其中F是权重k的全纯尖点形式，隐含常数现在取决于k。
On sup-norms of cusp forms of powerful level
设f是N阶L^2-正规化Hecke-Maass尖点新形式，Laplace特征值λ.证明了对于任意ε>0，|f| _ \infty <<_{λ，ε} N^{-1/12 + ε}.在N不能被“小平方”整除的情况下，指数得到进一步改进。我们的工作扩展和推广以前已知的结果在特殊情况下的N平方。
Type II$_1$ von Neumann algebra representations of Hecke operators on Maass forms and the Ramanujan-Petersson conjectures
经典的Hecke算子的马斯形式是unitarely等价的，直到一个交换阶段，完全积极的地图II $_1$因素，与一对同构的子因子，和一个交织酉。该表示是通过Hecke算子的量化表示获得的。在这种表示中，Hecke算子作用于Berezin量子化，即上半平面中$\PSL(2,\Z)$基本域的变形代数。Hecke算子继承了它们所作用的非交换代数，即矩阵不等式的丰富结构。利用这种构造，我们得到了对于每一个素数$p$，经典Hecke算子$T_p$的本质谱包含在Ramanujan Petersson定理所预言的区间$[-2\sqrt p, 2\sqrt p]$中.特别是，给定一个包含$[-2\sqrt p, 2\sqrt p]$的开区间，最多有有限个可能的例外特征值位于此区间之外。获得这种Hecke算子表示的主要工具（unitarely等价于经典表示，直到交换阶段）是Schurr型，$\PGL(2,\Q)$上状态的正“平方根“，通过$\PGL(2,\Q)$中的平移测量基本域的位移。“平方根“是从限制于$\PGL(2, \Q)$的$\PSL(2,\R)$的离散级数表示的矩阵系数获得的。本文的方法也适用于$\PSL(2,\Z)$的任意有限指数模子群$Γ_0(p^n)$，$n\geq 1$.在这种情况下，Hecke算子的本质范数等于陪集Hilbert空间$\ell^2((Γ_0(p^n))\back…
Blomer and Maga recently proved that, if $F$ is an $L^2$-normalized Hecke Maass cusp form for $\mathrm{SL}_n(\mathbb Z)$, and $Ω$ is a compact subset of $\mathrm{PGL}_n(\mathbb R)/\mathrm{PO}_n(\mathbb R)$, then we have $\|F|_Ω\|_\infty\ll_Ωλ_F^{n(n-1)/8-δ_n}$ for some $δ_n>0$, where $λ_F$ is the Laplacian eigenvalue of $F$. In the present paper, we prove an explicit version of their result.
Bounds for short character sums for $GL(2) \times GL(3)$ twists
本文研究 GL(2) 与 GL(3) 自守形式经非平凡模素数特征扭转后的短特征和，证明了一个含 N、p 与 ε 的新上界。
Let $π$ be a Hecke--Maass cusp form for $\rm SL_3(\mathbb{Z})$ with normalized Hecke eigenvalues $λ_π(n,r)$. Let $f$ be a holomorphic or Maass cusp form for $\rm SL_2(\mathbb{Z})$ with normalized Hecke eigenvalues $λ_f(n)$. In this paper, we are concerned with obtaining nontrivial estimates for the sum \begin{equation*} \sum_{r,n\geq 1}λ_π(n,r)λ_f(n)e\left(t\,φ(r^2n/N)\right)V\left(r^2n/N\right), \end{equation*} where $e(x)=e^{2πix}$, $V(x)\in \mathcal{C}_c^{\infty}(0,\infty)$, $t\geq 1$ is a large parameter and $φ(x)$ is some real-valued smooth function. As applications, we give an improved s…
The sup-norm problem for GL(2) over number fields
本文解决了数域上 GL(2) 群无平方因子水平球面 Hecke-Maass 新形式的上确界范数问题，在特征值和水平方面实现幂次节省，其界具有 Weyl 型指数并改进了已知特例，在全实域上与有理数域最佳混合结果相当。
Hybrid bounds for the sup-norm of automorphic forms in higher rank
本文针对高阶紧商空间上的 Hecke-Maass 形式，给出了关于特征值与判别式一致成立的显式次凸混合上确界范数界，并推广到任意奇数次可除代数的 Eichler 型序情形。
On the global sup-norm of GL(3) cusp forms
本文研究了非紧空间上GL(3)尖点形式的全局上确界范数问题，通过建立关于Laplace特征值的逐点上界，证明了tempered形式的全局上确界范数界，并推导出GL(3) Jacquet-Whittaker函数的新一致上界。
On the $L^2$ restriction norm of large level
该论文证明了大无平方因子级全纯Hecke新形式在垂直测地线段上的质量等分布定理，关联了其$L^2$质量与扭曲L-函数二阶矩。
The sup-norm problem for newforms of large level on $\operatorname{PGL}(n)$
本文研究了大水平新形式在PGL(n)上的上范数问题，通过新的降维理论和Atkin-Lehner算子推广，证明了次下界并基于GRH条件扩展结果。
The sup-norm problem in analytic number theory asks for the largest value taken by a given automorphic form. We observe that the function-field version of this problem can be reduced to the geometric problem of finding the largest dimension of the $i$th stalk cohomology group of a given Hecke eigensheaf at any point. This problem, in turn, can be reduced to the intersection-theoretic problem of bounding the "polar multiplicities" of the characteristic cycle of the Hecke eigensheaf, which in known cases is the nilpotent cone of the moduli space of Higgs bundles. We solve this problem for newfor…
Restriction of Hecke eigenforms to horocycles
我们证明了一个尖锐的上界上的$L^2$ -范数的Hecke本征形限制在一个horocycle，作为重量趋于无穷大。
$L^q$-norm bounds for arithmetic eigenfunctions via microlocal Kakeya-Nikodym estimate
本文通过微局Kakeya-Nikodym估计改进了算术特征函数的$L^6$-norm界，利用微局分解和算术放大技术获得更优的全局界。
On the zeta function on the line Re(s) = 1
We show the estimates \inf_T \int_T^{T+δ} |ζ(1+it)|^{-1} dt =e^{-γ}/4 δ^2+ O(δ^4) and \inf_T \int_T^{T+δ} |ζ(1+it)| dt =e^{-γ} π^2/24 δ^2+ O(δ^4) as well as corresponding results for sup-norm, L^p-norm and other zeta-functions such as the Dirichlet L-functions and certain Rankin-Selberg L-functions. This improves on previous work of Balasubramanian and Ramachandra for small values of δand we remark that it implies that the zeta-function is not universal on the line Re(s)=1. We also use recent results of Holowinsky (for Maass wave forms) and Taylor et al. (Sato-Tate for holomorphic cusp forms )…
In [BS07] Breuil and Schneider formulated a conjecture on the equivalence of the existence of invariant norms on certain $p$-adically locally algebraic representations of $GL_n(F)$ and the existence of certain de-Rham representations of $Gal(\bar{F}/F)$, where $F$ is a finite extension of $\mathbb{Q}_p$. In [Bre03b, DI13] Breuil and de Ieso proved that in the case $n = 2$ and under some restrictions, the existence of certain admissible filtrations on the $ϕ$-module associated to the two-dimensional de-Rham representation of $Gal(\bar{F}/F)$ implies the existence of invariant norms on the corre…
Restriction of toral eigenfunctions to hypersurfaces and nodal sets
本文给出了三维标准平坦环面上拉普拉斯算子的特征函数对曲率非零曲面的限制的L^2范数的一致上界和下界。我们还提出了几个相关的结果，本征函数的节点集。
Let f: S' --> S be a finite and faithfully flat morphism of locally noetherian schemes of constant rank n and let G be a smooth, commutative and quasi-projective S-group scheme with connected fibers. For every r>0, let Res_{G}^{(r)}: H^{r}(S_{et},G)---> H^{r}(S'_{et},G) and Cores_{G}^{(r)}: H^{r}(S'_{et},G)---> H^{r}(S_{et},G) be, respectively, the restriction and corestriction maps in etale cohomology induced by f. For certain pairs (f, G), we construct maps α_{r}: Ker Cores_{G}^{(r)}---> Coker Res_{G}^{(r)} and β_{r}: Coker Res_{G}^{(r)}---> Ker Cores_{G}^{(r)} such that α_{r}oβ_{r}=β_{r}oα_…
Universal quadratic forms and elements of small norm in real quadratic fields
对任意正整数M，证明了有无穷多个实二次域不存在M元泛二次型（对它们的交叉系数的奇偶性没有任何限制）.
On the square-free representation function of a norm form and nilsequences
我们表明，限制平方的代表功能附加到一个规范形式的免费号码不相关的nilsequences。通过将这个结果与Browning和作者以前的工作相结合，我们得到了Harpaz和Wittenberg最近关于有理点的纤维化方法的工作中所使用的应用。
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 …
We estimate the $L^2$ norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace-Beltrami operator on the standard flat torus $\mathbb{T}^d$, $d\ge2$. We reduce getting correct bounds to counting lattice points in the intersection of some $ν$-transverse bands on the sphere. Moreover, we prove the correct bounds for rational totally geodesic submanifolds of arbitrary codimension. In particular, we verify the conjecture of Bourgain-Rudnick on $L^2$-restriction estimates for rational hyperplanes. On $\mathbb{T}^2$, we prove the uniform $L^2$ restriction bounds …
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