Related papers: On Poincar\'e series of half-integral weight

Let $ \Gamma $ be a congruence subgroup of $ \mathrm{Sp}_{2n}(\mathbb Z) $. Using Poincar\'e series of $ K $-finite matrix coefficients of integrable discrete series representations of $ \mathrm{Sp}_{2n}(\mathbb R) $, we construct a…

Using Poincar\'e series of $ K $-finite matrix coefficients of integrable antiholomorphic discrete series representations of $ \mathrm{Sp}_{2n}(\mathbb R) $, we construct a spanning set for the space $ S_\rho(\Gamma) $ of Siegel cusp forms…

Let $ G $ be a connected semisimple Lie group with finite center. We prove a formula for the inner product of two cuspidal automorphic forms on $ G $ that are given by Poincar\'e series of $ K $-finite matrix coefficients of an integrable…

Let $\G\subset \mathrm{SL}_{2}(\R)$ be a cofinite Fuchsian subgroup, and let $i\infty$ be a cusp of $\G$. For $k\in\Z_{\geq 0}$, let $\Sk$ denote the complex vector space of cusp forms of weight-$k$, with respect to the Fuchsian subgroup…

Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix…

Let $\Gamma\subset \textrm{PSL}_2({\mathbb R})$ be a Fuchsian group of the first kind having a fundamental domain with a finite hyperbolic area, and let $\widetilde\Gamma$ be its cover in $\textrm{SL}_2({\mathbb R})$. Consider the space of…

We calculate the Jacobi Eisenstein series of weight $k \ge 3$ for a certain representation of the Jacobi group, and evaluate these at $z = 0$ to give coefficient formulas for a family of modular forms $Q_{k,m,\beta}$ of weight $k \ge 5/2$…

In this paper, we study the $ K $-finite matrix coefficients of integrable representations of the metaplectic cover of $ \mathrm{SL}_2(\mathbb R) $ and give a result on the non-vanishing of their Poincar\'{e} series. We do this by adapting…

Let $\Gamma\subset PSL_2(\mathbb{R})$ be a Fuchsian group of the first kind which has a cusp $i\infty$ of width one. In this paper, we first consider a generating function formed with the Niebur--Poincar\'e series $\{F_{m,s}(\tau)\}_{m\ge…

We prove a strengthening of Mui\'c's integral non-vanishing criterion for Poincar\'e series on unimodular locally compact Hausdorff groups and use it to prove a result on non-vanishing of L-functions associated to cusp forms of…

We derive new Poincar\'e-series representations for infinite families of non-holomorphic modular invariant functions that include modular graph forms as they appear in the low-energy expansion of closed-string scattering amplitudes at genus…

We show, for levels of the form $N = p^a q^b N'$ with $N'$ squarefree, that in weights $k \geq 4$ every cusp form $f \in \mathcal{S}_k(N)$ is a linear combination of products of certain Eisenstein series of lower weight. In weight $k=2$ we…

In this article, for $n\geq 2$, we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of $\mathrm{SU}\big((n,1),\mathbb{C}\big)$.…

We prove the conjectures on the ($L^{\infty}$)-sizes of the spaces of Siegel cusp forms of degree $n$, weight $k$, for any congruence subgroup in the weight aspect as well as for all principal congruence subgroups in the level aspect, in…

In his Inventiones papers in 1995 and 1998, Borcherds constructed holomorphic automorphic forms $\Psi(F)$ with product expansions on bounded domains $D$ associated to rational quadratic spaces $V$ of signature (n,2). The input $F$ for his…

Let $ G $ be a real simple linear connected Lie group of real rank one. Then, $ X := G/K $ is a Riemannian symmetric space with strictly negative sectional curvature. By the classification of these spaces, $X$ is a real/complex/quaternionic…

Let $\Gamma$ be a geometrically finite Fuchsian group and suppose that $\chi\colon\Gamma\to\mathrm{GL}(V)$ is a finite-dimensional representation with non-expanding cusp monodromy. We show that the parabolic Eisenstein series for $\Gamma$…

We compute explicit formulae for Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of potent and saturable principal congruence subgroups of $\mathrm{SL}_4^m(\mathfrak{o})$ ($m\in\mathbb{N}$)…

Let G be a semisimple Lie group with no compact factors, K a maximal compact subgroup of G, and $\Gamma$ a lattice in G. We study automorphic forms for $\Gamma$ if G is of real rank one with some additional assumptions, using dynamical…

A holomorphic discrete series representation $(L_\pi,H_\pi)$ of a connected semi-simple real Lie group $G$ is associated with an irreducible representation $(\pi,V_{\pi})$ of its maximal compact subgroup $K$. The underlying space $H_\pi$…