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I am not being able to understand what a cusp form is. The book i am following iwaniec kowalski analytic number theory "holomorphic modular forms" chapter . in particular what is a cusp exactly. some visulization would help.And why is it so important in number theory?

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    $\begingroup$ Did you try to google "cusp form"? Read Wikipedia article? $\endgroup$ Commented Nov 28, 2013 at 19:06

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A cusp form is a modular form which vanishes at the cusps.

As for visualization, well... below, you'll find a rendering of the real part of the simplest cusp form of all, the modular discriminant $\Delta(q)$. Beautiful, but singularly unhelpful as far as intuition goes.

And as for the question "why are they important in number theory?", it is a very broad question. You'll have to do some reading yourself. Similar questions have been asked here before, such as this one.

enter image description here

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