I learned two pieces of info on the relations of poles and zeros of a modular form $f$ as follows: $\mathbb{H}$ is the upper half plane, $G$ is the modular group $\text{SL}_2(\mathbb{Z})$, $\omega=\frac{1}{2}+\frac{\sqrt{3}i}{2}$, $k$ is the weight of $f$ and $f$ is not identically zero, $v_p(f)$ denotes the multiplicity of $f$ at the zero or pole point $p$. One fomula is \begin{align} v_{\infty}(f)+\frac{1}{2}v_i(f)+\frac{1}{3}v_{\omega}(f)+\sum_{i,\omega\neq p\in \mathbb{H}/G}v_p(f)=\frac{k}{6}. \end{align} And another states that the number of zeros and the number of poles of $f$ are equal (counting multiplicity), since $f$ is a meromorphic function on the Riemann surface, the closure of $\mathbb{H}/G$. Are there any relations between these two statements? Or are the two statements correct? Some guidance would help.
Are there any relations between the two statements about the poles and zeros of a modular form?
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number-theory
complex-analysis
modular-forms
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1[tag:modular-arithmetic] is **definitely** inappropriate here... :) – 2011-08-07
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0@J.M.:Thanks for reminding.:) – 2011-08-07