# Are there any relations between the two statements about the poles and zeros of a modular form?

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.

## 1 Answer

Both statements are correct. Let $\mathbb H^*=\mathbb H \cup \{\infty\}$. Consider the quotient map $$p: \mathbb H^* \rightarrow \mathbb H^*/G$$

Your first statement that: \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} is a statement concerning meromorphic modular forms on $\mathbb H^*$, rather than on $\mathbb H^*/G$.

Your second statement that a meromorphic function on a compact Riemann surface has an equal number of zeros and poles is a statement that applies to functions on $\mathbb H^*/G$, rather than on $\mathbb H^*$.

Here is how they fit together and are mutually consistent: First of all, a meromorphic function on $\mathbb H^*/G$ lifts to a meromorphic modular form of weight zero on $\mathbb H^*$, thus we can take $k=0$ in your first statement.

Now all that remains is to explain how the $\frac{1}{2}$ and $\frac{1}{3}$ factors come into play. These arise from the complex structure that we put on $\mathbb H^*/G$ (see Milne's notes on modular forms, http://www.jmilne.org/math/CourseNotes/mf.html, pg. 31-32, for details).

Briefly, the quotient map at the points $i$ and $\omega$ is ramified, so it locally looks like $z \mapsto z^2$ and $z \mapsto z^3$ respectively. Thus, for the meromorphic function on $\mathbb H^*/G$, a pole or zero of multiplicity $\nu$ corresponds to a pole or zero of multiplicity $2\nu$ or $3\nu$ on the modular function on $\mathbb H^*$.

• A great answer! Thank you very much! – user14242 Aug 7 '11 at 14:49