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.


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 $\mathbb H^*$.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.