Related papers: Eisenstein series on affine Kac-Moody groups over …

Let $G$ be a representation-theoretic Kac--Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We first consider Kac-Moody analogs of Borel Eisenstein series (induced from quasicharacters on the Borel), and…

We carry out "Hecke summation" for the classical Eisenstein series $E_k$ in an adelic setting. The connection between classical and adelic functions is made by explicit calculations of local and global intertwining operators and Whittaker…

Generalised Eisenstein series are non-holomorphic modular invariant functions of a complex variable, $\tau$, subject to a particular inhomogeneous Laplace eigenvalue equation on the hyperbolic upper-half $\tau$-plane. Two infinite classes…

Let G be a reductive algebraic group over a local field K or a global field F. It is well know that there exists a non-trivial and interesting representation theory of the group G(K) as well as the theory of automorphic forms on the…

We generalize the work of Ohta on the congruence modules attached to elliptic Eisenstein series to the setting of Hilbert modular forms. Our work involves three parts. In the first part, we construct Eisenstein series adelically and compute…

This paper presents a very simple explicit description of Langlands Eisenstein series for ${\rm SL}(n,\mathbb Z)$. The functional equations of these Eisenstein series are heuristically derived from the functional equations of certain…

This paper provides an introduction to Eisenstein measures, a powerful tool for constructing certain $p$-adic $L$-functions. First seen in Serre's realization of $p$-adic Dedekind zeta functions associated to totally real fields, Eisenstein…

In this paper we investigate a result of Ueno on the modularity of generating series associated to the zeta functions of binary Hermitian forms previously studied by Elstrodt et al. We improve his result by showing that the generating…

Motivated by string theory scattering amplitudes that are invariant under a discrete U-duality, we study Fourier coefficients of Eisenstein series on Kac-Moody groups. In particular, we analyse the Eisenstein series on $E_9(R)$, $E_{10}(R)$…

From the theory of modular forms, there are exactly $[(k-2)/6]$ linear relations among the Eisenstein series $E_k$ and its products $E_{2i}E_{k-2i}\ (2\le i \le [k/4])$. We present explicit formulas among these modular forms based on the…

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…

In this work, we define a new type of Eisenstein-like series by using Pell-Lucas numbers and call them the Pell-Lucas-Eisenstein Series. Firstly, we show that the Pell-Lucas-Eisenstein series are convergent on their domain. Afterwards we…

Let $p$ be a prime $\ge 5$. We establish explicit rates of overconvergence for members of the "Eisenstein family", notably for the $p$-adic modular function $V(E_{(1,0)}^{\ast})/E_{(1,0)}^{\ast}$ ($V$ the $p$-adic Frobenius operator) that…

We prove a version of the Gindikin-Karpelevich formula for untwisted affine Kac-Moody groups over a local field of positive characteristic. The proof is geometric and it is based on the results of [1] about intersection cohomology of…

We prove the quantum ergodicity of Eisenstein series on the arithmetic hyperbolic 3-manifold $\operatorname{PSL}_2(\mathcal{O}_F)\backslash \mathbb{H}^3$, where $F$ is an imaginary quadratic field with ring of integers $\mathcal{O}_F$ and…

We form real-analytic Eisenstein series twisted by Manin's noncommutative modular symbols. After developing their basic properties, these series are shown to have meromorphic continuations to the entire complex plane and satisfy functional…

In this paper, based on Serre's $p$-adic family of Eisenstein series, we prove a general family of congruences for Eisenstein series $G_k$ in the form $$ \sum_{i=1}^n g_i(p)G_{f_i(p)}\equiv g_0(p)\mod p^N, $$ where…

Consider a subgroup of finite index of modular group. We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian of the corresponding modular curve. By BelyI theorem, such a criterion would apply to any curve over a…

We identify a class of "semi-modular" forms invariant on special subgroups of $GL_2(\mathbb Z)$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an…

We establish a Siegel-Weil formula for classical groups over a function field with odd characteristic, which asserts in many cases that the Siegel Eisenstein series is equal to an integral of a theta function. This is a function-field…