TITLE: Modular forms of fractional weight
QUESTION [27 upvotes]: Modular forms of integral weight are prominent in number theory.
Furthermore, there are $\theta$-functions and the $\eta$-function, having weight 1/2,
which also have a rich theory. 
But I have never seen a modular form of weight e.g. 1/3. 
I have been wondering about this for a long time. Are there
examples of modular forms of fractional weights other than multiples of 1/2?
And if yes, is there are reason why they are poorly studied?

REPLY [16 votes]: Automorphic forms naturally live on the adelic points of a reductive algebraic group (modulo rational points and a compact subgroup giving the level). One may interpret automorphic forms
of fractional weights to be automorphic forms which live on a topological cover of the adelic group as in the classical metaplectic works mentioned earlier. In this vein, the most modern treatment is in the paper of Brylinski-Deligne. The only arithmetically related work seems to be due to Marty Weissman. The Brylinski-Deligne paper works equally well over function fields and it would be interesting to see its connections with Lafforgue's works.