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I am self studying Analytic number theory from Apostol Dirichlet series and modular functions in number theory and I have a doubt regarding defintions of modular forms and modular functions. My question is - Both in definitions of Modular functions and entire Modular forms, the function f is assumed to be meromorphic and analytic in upper half plane H respectively . Is there any specific reason that functions are assumed to behave particular in upper half plane H not any other region of complex plane? Why only upper half plane H is chosen not any other region?

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  • $\begingroup$ they are only defined on $H$; moreover they usually have the real line as a natural boundary. $\endgroup$ – Lord Shark the Unknown Jul 21 at 19:38
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    $\begingroup$ Consider a modular form $f$ on $SL(2, \mathbb{Z})$, it satisfies $f(z) = f(z+1)$. Therefore, it has a Fourier expansion $f(z) = \sum_{n=0}^{\infty} a_n e^{2\pi inz} \ (z \in \mathcal{H})$, where there are no terms of $n <0$ because of the growth conditions we have imposed in the definiton. Note $|e^{2\pi iz}|= e^{-2 \pi Im(z)}$. What would happen if $Im(z) < 0$? $\endgroup$ – Parthiv Basu Jul 21 at 20:51
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A naïve answer would be to take the sum and the RHS modulo $1$; however, a "number" $\mathfrak{a}$ modulo any number $n$ is in fact an equivalence class $$\mathfrak{a}=[a]_{n}=\{b: a\sim_n b\},$$ where $\sim_n$ is the equivalence relation defined by $a\sim_n b\iff a=b+kn$ for some $k\in \Bbb Z$. Then $$[\color{blue}{1}]_1\color{green}{+_1}[\color{red}{1}]_1\color{green}{:=}[\color{blue}{1}+\color{red}{1}]_1=[0]_1=[3]_1$$ because $1+1=0+(2\times 1)$ and $3=0+(3\times 1)$.

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