Let $ψ$ be a smooth compactly supported function on $\mathbb{X} = SL(2,\mathbb{Z})\backslash\mathbb{H}$. In this paper, we are interested in the joint cubic moments of automorphic forms when the spectral parameters go to infinity. We show that the diagonal case for Eisenstein series $\int_{\mathbb{X}}ψ(z)E(z,1/2+it)^{3} dμz = \mathcal{O}_ψ(t^{-1/3+\varepsilon})$. In off-diagonal case we prove $\frac{1}{2\log t}\int_{\mathbb{X}}ψ(z)|E(z,1/2+it)|^{2}g(z)dμz = o(1)$ as long as $\min\{t , t_{g}\} \rightarrow \infty$. Finally we show
$\int_{\mathbb{X}}ψ(z)f^{2}(z)g(z)dμz = o(1)$ in the range $|t_{f} - t_{g}| \leq t_{f}^{2/3-\varepsilon}$ where $f,g$ are two Hecke-Maass cusp forms.
In this paper, we explicitly construct mock modular forms whose shadows are Eisenstein series of arbitrary integral and half-integral weight, level and character at the cusps $\infty$ and $0$. As an application, we give explicit construction of harmonic weak Maass forms which are Hecke eigenforms and are the preimages of $Θ^k, k \in \{ 3, 5, 7\}$ under the shadow operator, where $Θ$ is the classical Jacobi theta function.
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.
We introduce a new trace formula of Kuznetsov type involving the central standard L-values and the Whittaker periods of cuspidal automorphic representations of PGL_n(Q) which are spherical at the archimedean place. As an application, we show a simultaneous non-vanishing of standard L-values at n-1 points on the critical strip for infinitely many Hecke-Maass cuspidal newforms.
We show that Landau-Siegel zeros for Dirichlet $L$-functions do not exist or quantum unique ergodicity for $\mathrm{GL}_2$ Hecke-Maass newforms holds with an effective rate of convergence. This follows from a more general result: Landau-Siegel zeros of Dirichlet $L$-functions repel the zeros of all other automorphic $L$-functions from the line $\mathrm{Re}(s)=1$.
We derive the Fourier expansion of scalar-valued Eisenstein series for O(2, n+2) using classical methods of Siegel, Braun, Zagier, Bruinier and others. We assume that the underlying lattice splits two hyperbolic planes. Finally we prove for the Eisenstein series at the standard cusp that they belong to the Maass space for O(2, n + 2), an analogue of the "Spezialschar" for Siegel modular forms introduced by Maass, at least at all local places p, where the localization of the underlying lattice is maximal.
p-adic Eisenstein-Kronecker series and non-critical values of p-adic Hecke L-functions of an Imaginary Quadratic Field when the conductor is divisible by p
We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group $\mathrm{SL}_3(\mathbb Z)$. When the underlying form is self-dual, we show that there are $\gg_\varepsilon X^{5/6-\varepsilon}$ sign changes among the coefficients $\{A(m,1)\}_{m\leq X}$ and that there is a positive proportion of sign changes for many self-dual forms. Similar result concerning the positive proportion of sign changes also hold for the real-valued coefficients $A(m,m)$ for generic $\mathrm{GL}_3$ cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of $\…
On the Ramification of Hecke Algebras at Eisenstein Primes
利用Wiles的模技巧，研究了权为2、素级为N的Hecke代数在Eisenstein素数上的局部化.在此基础上，我们从形变理论的角度恢复了Mazur的一些结果（“模曲线和Eisenstein理想”）。结合Merel的一个定理，我们得到了Q（N^（1/p））的p-部分的新信息，其中p和N是素数，N = 1 mod p.
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…
Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight $k$ and level $Γ_0(\ell)$ at the maximal Eisenstein ideal containing $p$. We give a necessary and sufficient condition for the $\mathbb{Z}_p$-rank of this Hecke algebra to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for …
We construct a family of harmonic Maass forms of polynomial growth of any level corresponding to any cusp whose shadows are Eisenstein series of integral weight. We further consider Dirichlet series attached to a harmonic Maass form of polynomial growth, study its analytic properties, and prove an analogue of Weil's converse theorem.
