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?!