# Modular form with pole of infinite order

In the definition of a (weakly holomorphic) modular form we require a specific growth behaviour at all the cusps. I assume that this requirement is not void, i.e. not automatically satisfied.

However, I have never seen an example of a modular form, which satisfies the growth condition at one cusp but not at another.

So my question is: Can somebody give an example of a function which transforms like a modular form (I do not care about the weight or the congruence subgroup) and which is weakly holomorphic (i.e. has at most a pole) a some cusp but not at some other cusp (i.e. has an essential singularity).

Are there also examples which are interesting from the point of arithmetic geometry/number theory/...

EDIT: Does somebody also know an example where the weight is non-zero

• Ok, when I was thinking about this, I also had in mind just taking exp but I this didn't work because it wouldn't transform correctly. Of course it does transform correctly, as shown by David and Paul in weight 0. Thanks so far. Does somebody know a less trivial example, in non-zero weight. – wood Jun 21 '11 at 13:01

Let $\Gamma = \Gamma_0(2)$, and let $\Delta$ be the usual weight 12 cusp form. Then $f(z) = \Delta(2z) / \Delta(z)$ is a meromorphic modular function of weight 0 and level $\Gamma$, holomorphic on the upper half-plane, and with a zero at the cusp $\infty$ and a pole at the cusp $0$. So $\exp(f(z))$ is an example of a holomorphic function on the upper half-plane which transforms like a modular form of weight 0 and level $\Gamma$, is holomorphic at $\infty$, and has an essential singularity at $0$.
This is a standard riff: let $E$ be a weight-12 holomorphi Eisenstein series vanishing at all but one cusp for some congruence subgroup of $SL(2,Z)$. Let $f$ be the level-one cuspform of weight 12, which we know does not vanish in the interior. Then $exp(E/f)$ is weight-0, holomorphic in the interior and at all cusps but the one where $E$ is non-vanishing, at which it has an essential singularity.