# Modular forms question

Can someone enlighten me with the question in the next page: http://www.physicsforums.com/showthread.php?p=3208664#post3208664

I am asked to find all the modular forms with weight $k$ which don't have zeros on the upper half plane.

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Would you mind retyping the question here? –  user641 Mar 28 '11 at 10:52
I am asked to find all the modular forms with weight k which don't have zeros on the upper half plane. –  MathematicalPhysicist Mar 28 '11 at 10:58

The discriminant cusp form $\Delta$ is such a modular form, and its weight is twelve. It has a simple zero at infinity and no other zeroes. Suppose f is any modular form without zeroes in H. If f has a zero of order k at infinity, then $f/\Delta^k$ is a modular form with no zeroes or poles so it is constant. In particular the weight of f is $12k$.
Wait a minute the weight of $\delta$ is 12 so the weight of $\delta ^k$ is 12k, and the weight of $\frac{f}{\delta^k}$ should be -12k, right? –  MathematicalPhysicist Mar 28 '11 at 13:35
@MathematicalPhysicist: Dan proves (very succinctly) that the only modular forms with no zeros on the upper half plane are $c \Delta^k$ where c is any constant. (If such a modular form f has weight 12k, then dividing f by $\Delta^k$ gives you a modular form of weight 0, so it must be a constant. $f/\Delta^k=c$, so $f=c\Delta^k$.) –  Jonas Kibelbek Mar 28 '11 at 17:08