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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.

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    [tag:modular-arithmetic] is **definitely** inappropriate here... :)2011-08-07
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    @J.M.:Thanks for reminding.:)2011-08-07

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