TITLE: Why only half-integral weight automorphic forms?
QUESTION [18 upvotes]: Why is that the theory of automorphic forms concentrates on the case of half-integral weight? I read in Borel's book "Automorphic forms on $SL_2$" (Section 18.5) that by considering the finite or universal coverings of $SL_2$, one can define modular forms with rational or even real weight. Borel mentions in passing that the case of half-integral weight is particularly important. Why?!

REPLY [2 votes]: You may want to look at the following article:
MR1809555 (2002b:11056) 
Ibukiyama, T.
Modular forms of rational weights and modular varieties.
Abh. Math. Sem. Univ. Hamburg 70 (2000), 315–339.