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?
Best Answer
I am no expert here, but I believe modular forms of fractional weight (e.g. of weight 1/3) appear more naturally as forms on metaplectic covers of GL(2) (e.g. on the cubic cover) and over fields containing the relevant roots of unity (e.g. the third roots of unity). Kubota around 1970 initiated the study of these covers, and a few years later Patterson initiated the study of the forms on them. Patterson's two papers here seem to be a good starting point. Later Patterson alone and jointly with Heath-Brown applied the new knowledge to old objects in number theory like Gauss and Kummer sums, see e.g. here and here. Patterson and Kazhdan in 1984 greatly generalized Kubota's work to metaplectic covers of GL(r), see here.
All in all I believe the general theory is technically quite involved which explains why so few are familiar with it. However, forms of fractional weight are no doubt an organic part of number theory, but they appear more naturally on symmetric spaces of higher rank.