Are these two functions equal?

When are two modular functions equal?

• Suppose $f, g \in M_{k}(\Gamma)$ for some $\Gamma$ a congruence subgroup of $PSL_{2}(\mathbb{Z})$. If $k \geq 1$ then it is a finite computation to determine whether or not $f = g$. What if $k = 0$, that is, if $f$ and $g$ are both modular functions and I know their $q$-expansions, how can I tell when $f = g$?

• Answer:

  As Qiaochu has pointed out, the answer to this question depends on what exactly you mean by "modular function". If you mean a function that satisfies the weight k functional equation for the action of $\Gamma$ and is holomorphic everywhere on the upper half-plane and the cusps, then this is what is generally called a modular form; and the only weight 0 modular forms are constants, so the question is trivial, and for weight $k \ge 0$ I gave an answer http://math.stackexchange.com/questions/48815/agreement-of-q-expansion-of-modular-forms/4885. If you mean only that your functions are meromorphic on $\mathcal{H}^* = \mathcal{H} \cup \{\text{cusps}\}$, then (whatever $k$ is) you can't answer the question just knowing the $q$-expansion, but you can if you know more about the functions involved. If you know some finite set of points of $\Gamma \backslash \mathcal{H}^*$ such that your functions have no poles away from these points, and you can bound the order of the poles at the bad points, then you're in good shape, since you can find a modular form $h$ of some weight $k$ such that $fh$ and $gh$ are modular forms, and that reduces you to the first case. (Alternatively, $f - g$ is a section of a line bundle on a compact Riemann surface, so if it vanishes at $\infty$ with multiplicity larger than the degree of the line bundle, it must be zero.)