Hi,


I am interested in the following fact:


Suppose that $f(z)$ is a modular function over $\text{SL}\_2(\mathbb{Z})$ such that it has the $q$-expansion


$f(z) = q^{-m} + \displaystyle \sum\_{n=1}^\infty a\_m(n) q^n$ 


It is claimed in Ono, K, "The partition function and Hecke operators", Advances in Mathematics 228 (2011), 527-534 that this function (which he denotes $j\_m(z)$) is unique.


I know that modularity imposes strict restrictions on what modular functions must look like, but I have a problem seeing this claim. Can anyone offer a short proof or some ideas as to why this is the case?


Thanks in advance.



Uniqueness of modular functions with a certain $q$-expansion
I *would* say that the basic fact underlying this is that the only regular functions on a compact Riemann surface are constant functions, but I'm guessing that this isn't the missing detail in your case.


A modular function for $\mathrm{SL}\_2(\mathbb{Z})$ is (basically by definition in your context) a holomorphic function on the quotient of the upper half-plane by its usual action of $\mathrm{SL}\_2(\mathbb{Z})$ by fractional linear transformations. At the end of the day, this amounts to a holomorphic function on the Riemann sphere minus the point $\infty$ (if you haven't done so already, reading about the modular function known as the $j$-invariant will help to explain this). The $q$-expansion of such a function is its local expansion about $\infty$. In particular, if two such functions have $q$-expansions of the type you specify, then their difference is regular everywhere and vanishes at $\infty$, and hence must be the constant zero.


