Modular form with pole of infinite order

Question

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

Answer 1

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 example isn't terribly interesting from a number-theoretical viewpoint; I don't know of any "interesting" examples that aren't meromorphic, and perhaps the point of imposing the meromorphicity condition is to slim down your space until it becomes small enough to be interesting. :-)

Answer 2

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.

Answer 3

If you take your favourite (holomorphic) modular form of nonzero weight k and level one, and multiply it by one of the examples in weight 0, you will get something that transforms as a weight k form, but has the same singularity behaviour as the weight 0 examples.