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Given a (meromorphic or holomorphic) modular form $f$ of weight $k$ on some genus zero congruence subgroup $\Gamma$, are there known bounds for the number of zeros and poles that $f$ has on the fundamental domain?

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I believe that a "modular forms" tag is very proper for a question about modular forms :-) –  Andrea Mori Jul 15 '11 at 16:39

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up vote 3 down vote accepted

For classical holomorphic modular forms $f$ of weight $k$ for the full modular group ${\rm SL}_2({\Bbb Z})$ there's a neat formula for the number of zeroes in the fundamental domain that can be obtained as an application of the Residue Theorem. The formula reads $$ v_\infty(f)+\frac12v_i(f)+\frac13v_\rho(f)+\sum_{z\notin\{\infty,i,\rho\}}v_z(f)=\frac k6, $$ where $v_P(f)$ denotes the order of vanishing at $P$, $\rho=e^{2\pi i/3}$ and the sum is extended over all the points in the fundamental domain. The formula is very basic and given in all textbooks about modular forms. If one reference should be given, I give Serre's Course d'Arithmetique (it's Theorem 3 in section 3.1 there).

For more general groups of modularity, such as the congruence groups $\Gamma_0(N)$ and similar, the standard approach to the problem of bounding the number of zeroes is to regard modular forms as holomorphic global sections of certain line bundles on the completed modular curves and apply the Riemann-Roch theorem. For instance, one gets at once that there are no non-trivial modular forms of negative weight of any level.

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I don't know, but I found this at http://mathoverflow.net/questions/23747/finding-zeroes-of-classical-modular-forms

"There are several papers which compute zeroes of modular forms for genus 0 congruence subgroups, such as "Zeros of some level 2 Eisenstein series" by Garthwaite et al published in Proc AMS and work of Shigezumi and others in levels 3, 5 and 7."

The MO question is actually about higher genus, but conceivably some of the discussion there could point you in the right direction.

Also, there's a brief discussion of zeros of modular forms on page 48 of Milne's notes, http://www.jmilne.org/math/CourseNotes/MF.pdf

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