If Г is a discrete Möbius group acting on the upper half-plane H of the complex plane, the quotient space H/Г is a Riemann surface R and the automorphic functions on Г correspond to meromorphic functions on R. If Г is a nondiscrete Möbius group acting on H, then H/Г is no longer a Riemann surface, and it is obvious that in this case there are no nonconstant automorphic functions on Г. The situation for automorphic forms is quite different. Automorphic forms of integral dimension for a discrete group Г correspond to meromorphic differentials on R, but even if Г is nondiscrete it may still support nontrivial automorphic forms. The problem of classifying those nondiscrete Möbius groups which act on H and which support nonconstant automorphic forms of arbitrary real dimension was raised and solved (rather indirectly) in [2] where, roughly speaking, function-theoretic methods are used to analyse all possible polynomial automorphic forms of integral dimension, and the results obtained then used to analyse the more general situation.
Beardon, A. F. Automorphic forms on nondiscrete Möbius groups. Glasgow mathematical journal, Tome 38 (1996) no. 2, pp. 249-253. doi: 10.1017/S0017089500031517
@article{10_1017_S0017089500031517,
author = {Beardon, A. F.},
title = {Automorphic forms on nondiscrete {M\"obius} groups},
journal = {Glasgow mathematical journal},
pages = {249--253},
year = {1996},
volume = {38},
number = {2},
doi = {10.1017/S0017089500031517},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500031517/}
}
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AU - Beardon, A. F.
TI - Automorphic forms on nondiscrete Möbius groups
JO - Glasgow mathematical journal
PY - 1996
SP - 249
EP - 253
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UR - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500031517/
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