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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 is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If $gamma$ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If $gamma in Gamma$ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.
We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the (rational) adel
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 functions. In
We identify the quantum limits of scattering states for the modular surface. This is obtained through the study of quantum measures of non-holomorphic Eisenstein series away from the critical line. We provide a range of stability for the quantum unique ergodicity theorem of Luo and Sarnak.
In recent work, M. Just and the second author defined a class of semi-modular forms on $mathbb C$, in analogy with classical modular forms, that are half modular in a particular sense; and constructed families of such functions as Eisenstein-like ser
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 Eisenstein-like