# Tagged Questions

A modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group.

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### Cusps in the Compactification of Modular Curves $Y(\Gamma)$

I'm currently reading some of the geometric theory behind the theory of modular forms, Diamond and Shurman's book is my main reference. If $\Gamma$ is a congruence subgroup of $SL_2(\mathbb{Z})$, the ...
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### $f\in M_k (\Gamma_1(N)$ and $g(z)=f(dz)$, then $g\in M_k(\Gamma_1 (dN)$

Let $N,d\in\mathbb{N}$ and $\Gamma_1(N)=\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb{Z}):a\equiv d\equiv 1\mod N,~c\equiv 0\mod N\}$. I need a proof (references or your ideas) of: ...
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### Right cosets $\Gamma_1(N) \cap\alpha^{-1} \Gamma_1(N)\alpha$ in $\Gamma_1$

In the book Introduction to Elliptic Curves and Modular Forms from Neal Koblitz there is a certain step in the solution of an exercise that is left to the reader (IV.3, Problem 5 and solution). Since ...
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### Modular equation for a quotient of eta functions

In this article at page $4$ F. Beukers introduces a function $$y(\tau) = \frac{\eta^{8}(6\tau)\eta^{4}(\tau)}{\eta^{8}(2\tau)\eta^{4}(3\tau)}\tag{1}$$ where $\tau$ is a complex number with positive ...
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### Modular transformations of $\eta(\tau)$

Under a modular transformation the Dedekind $\eta$ function transforms as $$\eta(-1/\tau) = \sqrt{-i}\eta(\tau).\tag*{(*)}$$Siegel gives a proof in this paper here that uses complex analytic ...
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### How can we show the equality $[SL_2(\mathbb Z): \Gamma_0(N)]=N\prod_{p\mid N}\left(1+\frac1p\right)$?

If you want to read question directly, please go to down. Firstly, let's give the definitions of $\Gamma(N)$, $\Gamma_1(N)$ ve $\Gamma_0(N)$: \Gamma(N) : = \left \{\left[ \begin{array}{cc}a & b ...