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In every introduction about Modular Forms (on $SL_2(\mathbb{Z})$ and congruence subgroups) one reads the term 'cusps'. A Modular Form should be holomorphic in the cusps. Can anybody explain to me, what is so special about cusps? Why is it necessary to put 'holomorphic in the cusps' in the definition?

I have already read that that makes the space of Modular Forms of a given weight finite dimensional. But why? Are there other benefits?

Thanks. P.Sch

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    One benefit is that we can write a modular form $f(\tau)$ as a series $\sum_{n=0}^\infty a_nq^n$ where $q=e^{2\pi i \tau}$. The cusp condition means that there are no negative coefficients. The space $\mathrm{SL_2}(\mathbb Z)\backslash\mathbb H$ is not compact - it has a cusp (a kind of asymptote going off to infitnity), but can be compactified by adding a point at infinity. We would like our modular forms to be holomorphic on this compactified space. – Mathmo123 May 06 '16 at 09:18

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Mathmo123 already wrote a comment which is really an answer, but I just want to add some details.

For the case $SL (2,\mathbb {Z})$ the holomorphicity at the cusp, which is also sometimes written as a growth condition, is \emph{equivalent} to the fact we can write our modular form as a Fourier expansion with non negative indices. We use the fact that the modular form is holomorphic at the cusp when we prove the well known formula about the multiplicities of the zeroes of a modular form of weight $k$ on the full modular group $$\frac{k}{12}=\sum_{p\in\Gamma_1/\mathbb{H}}{mult_p (f)} +\frac{1}{2}mult_{i}(f) +\frac{1}{3}mult_{\omega}(f)+ mult_{\infty}(f),$$ where in the sum you take the non elliptic points. This formula, combined with a linear algebra argument, leads to the final dimensionality of the space of modular forms.

This is subsequent to the compactification of the quotient space of the action of $SL (2,\mathbb {Z})$ on $\mathbb{H}$. This is not a priori compact, as you can see observing the fondamental domain of this action. The compact space is obtained just adjoing a cusp (the orbit of $\mathbb{Q}\cup\{\infty\}$) and of course you need holomorphicity at infinity to be able to write $mult_{\infty} f$ in the above formula.

A good reference for these facts is the first paragraph of Don Zagier's chapter in the book "1-2-3 of Modular Forms".

Gabriele
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  • and a pole is not a zero with negative multiplicity ? (can it be extended to non-holomorphic modular forms) ? – reuns May 15 '16 at 22:15
  • I think so. You probably have just to create a suitable contour of integration in the fundamental domain to obtain a formula of this kind; but I don't think it leads to a finiteness argument as in the holomorphic case. – Gabriele May 16 '16 at 12:02
  • Late to the party, but as long as the function is meromorphic on the upper half plane and at the cusps, the valence formula given above works, using poles as negative zeros. (See, e.g., Serre, A Course in Arithmetic for the contour integration.)

    And you're right, the resulting spaces are not finite dimensional. There are tricks to look at "nice" subspaces though. A key example is insisting the only poles are at the cusps (sometimes called weakly-holomorphic modular forms). In the weight zero case, this is a polynomial algebra in the j-invariant.

     – Ben Nov 23 '21 at 14:19