Super Automorphic Forms on the Super Upper Half Plane

Automorphic Forms and Holomorphic Functions on the Upper Half-plane

We define a set of holomorphic functions in terms of the Hauptmodul of a quotient Riemann surface and prove that these functions are holomorphic on the upper half-plane. It is also shown that these functions are automorphic forms of weight k with respect to a Fuchsian group.

Martin Functions of Fuchsian Groups and Character Automorphic Subspaces of the Hardy Space in the Upper Half Plane

We establish exact conditions for non triviality of all subspaces of the standard Hardy space in the upper half plane, that consist of character automorphic functions with respect to the action of a discrete subgroup of $SL_2(\mathbb R)$. Such spaces are natural objects in the context of the spectral theory of almost periodic differential operators and in asymptotics of approximations by entire functions. A naive idea: it should be completely parallel to the celebrated Widom characterization for Hardy spaces on Riemann surfaces with a minor modification, namely, one has to substitute the Green…

n维双曲空间上Vahlen群的算术同余子群的广义解析自守形式

Generalized Analytic Automorphic Forms for some Arithmetic Congruence subgroups of the Vahlen group on the n-Dimensional Hyperbolic Space

本文研究了一类新的解析型向量值和Clifford代数值的一元和二元向量值自守形式。对于经典模群及其算术同余子群的超复推广，讨论了在$R^n$的上半空间上被Dirac算子零化的Eisenstein型和Poincaré型级数，以及更一般地被迭代Dirac算子零化的Eisenstein型和Poincaré型级数.特别是，我们介绍（多）单演模形式超复杂的推广的经典θ组。

Stability of the space of Automorphic Forms under Local Deformations of the Lattice

First we explain the concept of local deformation over a 'parameter' algebra P, in particular the notion of a P-lattice in a Lie group. Purpose of this article is to define the spaces of automorphic resp. cusp forms on the upper half plane H for a P- (!) lattice of SL(2, R) and to investigate their structure. It turns out that in almost all cases these spaces are free modules over the complexified P of rank equal to the dimension of the spaces of automorphic resp. cusp forms for the body, which is the associated ordinary lattice in SL(2, R) . In other words almost every automorphic resp. cusp …

A Satake type theorem for Super Automorphic forms

Aim of this article is a Satake type theorem for super automorphic forms on a complex bounded symmetric super domain B of rank 1 with respect to a lattice. This theorem - roughly speaking - says that for large weight k and all p from 1 to infinity (both including) a super automorphic form on B is a super cusp form if and only if its p-norm with respect to a certain measure on the quotient of B is finite. And so in particular all these Lp-spaces coincide! We will give a proof of this theorem using an unbounded realization of B and Fourier decomposition at the cusps of the quotient mapped to inf…