# Modular forms of fractional weight

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?

• Good question! I've wondered about this too. I think it has something to do with the metaplectic group being special (unlike the other covers of SL_2) but I don't know enough to understand why. – Alison Miller Jan 23 '11 at 21:49

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.

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.

• I appreciate the reference. Though I may have been the first to apply Brylinski-Deligne in this vein, others including Peter McNamara and Wen-Wei Li have recently used Brylinski-Deligne's framework for discussing metaplectic groups. It should also be said that many topological covers do not arise from Brylinski-Deligne's setup. Classical modular forms of weight $1/3$ do not arise, since $\QQ$ only contains two roots of unity. But other modular forms of weight $1/3$ arise from the B-D setup when the number field contains a prim. cube root of unity. – Marty Jan 24 '11 at 7:05
• Note that classical modular forms of non-half-integral weight are possible-- real and even complex weights are possible since the fundamental group of $SL_2(R)$ is $Z$ and there are holomorphic line-bundles associated to characters of $Z$. But my personal opinion is that these (modular forms of arbitrary non-half-integer, non-integer weight) are primarily objects of analytic curiosity. I think that the automorphic forms most closely connected to arithmetic (through Langlands-ish methods) are automorphic forms on groups that can arise from Brylinski and Deligne's framework. – Marty Jan 24 '11 at 7:11

Modular forms of weight 1/2 are actually quite prominent in geoemtry (I can't speak for number theory). For instance, the 2nd order theta functions (which encode information about points of order two on abelian varieties, for instance) are of weight 1/2. They give a natural and important map from a certain cover of the moduli space of abelian varieties (specifically $\mathcal{A}_g^{(2n,4n)}$) into projective space which is injective for $n\geq 2$. Here are a few reference for 2nd order theta functions:

Kummer varieties and the moduli spaces of abelian varieties - van Geemen and van der Geer

Igusa's book on Theta Functions

Mumford's Tata Lectures on Theta.

Grushevsky's survey of the Schottky Problem (lots of things on the Schottky problem involve 2nd order theta functions)

• ok, I mentioned theta functions, and I wanted to exclude halfintegral forms. I will edit the question – wood Jan 24 '11 at 4:35