Related papers: A note on zeros of Eisenstein series for genus zer…

Let $\GN\leq\SLR$ be a genus zero Fuchsian group of the first kind with $\infty$ as a cusp, and let $\Ek$ be the holomorphic Eisenstein series of weight $2k$ on $\GN$ that is nonvanishing at $\infty$ and vanishes at all the other cusps…

We locate all of the zeros of the Eisenstein series associated with the Fricke groups $\Gamma_0^{*}(2)$ and $\Gamma_0^{*}(3)$ in their fundamental domains by applying and expanding the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer…

We locate all but $O(\sqrt{k\log{k}})$ zeroes of the half integral weight Eisenstein series $E_\infty(z,k)$ of $\Gamma_0(4)$ for $k$ sufficiently large. To do this, we relate $E_\infty(z,k)$ to $\Gamma_0(4)$'s other Eisenstein series,…

We prove that if $k$ and $\ell$ are sufficiently large, then all the zeros of the weight $k+\ell$ cusp form $E_k(z) E_{\ell}(z) - E_{k+\ell}(z)$ in the standard fundamental domain lie on the boundary. We moreover find formulas for the…

The present paper provides the details omitted from the more concise study "On the zeros of Eisenstein series for $\Gamma_0^* (5)$ and $\Gamma_0^* (7)$." We locate almost all of the zeros of the Eisenstein series associated with the Fricke…

We locate the zeros of the modular forms $E_k^2(\tau) + E_{2k}(\tau), E_k^3(\tau) + E_{3k} (\tau),$ and $E_k(\tau)E_l(\tau) +E_{k+l}(\tau),$ where $E_k(\tau)$ is the Eisenstein series for the full modular group $\text{SL}_2(\mathbb{Z})$. By…

We examine the zeros of newform Eisenstein series $E_{\chi_1,\chi_2,k}(z)$ of weight $k$ on $\Gamma_0(q_1 q_2)$, where $\chi_1$ and $\chi_2$ are primitive characters modulo $q_1$ and $q_2$, respectively. We determine the location and…

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 locate almost all the zeros of the Eisenstein series associated with the Fricke groups of level 5 and 7 in their fundamental domains by applying and extending the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (1970). We also use…

The cohomology $H^*(\Gamma, E) $ of a torsion-free arithmetic subgroup $\Gamma$ of the special linear $\mathbb{Q}$-group $\mathsf{G} = SL_n$ may be interpreted in terms of the automorphic spectrum of $\Gamma$. Within this framework, there…

The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing…

In this paper we generalize a well-known isomorphism between the space of cusp forms of weight $k$ for a Fuchsian subgroup of the first kind $\Gamma \subset\mathrm{SL}_{2}(\mathbb{R})$ and the space of certain Maa{\ss} forms of weight $k$…

This article proposes a new approach to studying the spectral Eisenstein series of weight $k$ on a congruence subgroup of $\text{SL}_2(\mathbb{Z})$ using Hecke's theory of Eisenstein series for the principal congruence subgroups. Our method…

We research the location of the zeros of the Eisenstein series and the modular functions from the Hecke type Faber polynomials associated with the normalizers of congruence subgroups which are of genus zero and of level at most twelve. In…

We locate all of the zeros of certain Poincare series associated with the Fricke groups $\Gamma_0^*(2)$ and $\Gamma_0^*(3)$ in their fundamental domains by applying and extending the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer…

We consider a set of generators for the space of Eisenstein series of even weight $k$ for any congruence group $\Gamma$ and study the set of all of their zeros taken for $\Gamma(1)$-conjugates of $\Gamma$ in the standard fundamental domain…

Let $M_k^\sharp(N)$ be the space of weakly holomorphic modular forms for $\Gamma_0(N)$ that are holomorphic at all cusps except possibly at $\infty$. We study a canonical basis for $M_k^\sharp(2)$ and $M_k^\sharp(3)$ and prove that almost…

Let $\Gamma$ be the Fuchsian group of the first kind. For an even integer $m\ge 4$, we describe the space $H^{m/2}\left(\mathfrak R_\Gamma\right)$ of $m/2$--holomorphic differentials in terms of a subspace $S_m^H(\Gamma)$ of the space of…

Let $\Gamma$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $\mathbb H$, and let $M = \Gamma \backslash \mathbb H$ be the associated finite volume hyperbolic Riemann surface. If $\gamma$ is parabolic, there…

Let $G$ be an affine or hyperbolic rank 2 Kac--Moody group over a finite field $\mathbb F_q$. Let $X=X_{q+1}$ be the Tits building of $G$, the $(q+1)$--homogeneous tree, and let $\Gamma$ be a non-uniform lattice in $G$. When $\Gamma$ is a…