# Modular Form Holomorphic at Cusps

Let $f$ be a weight $k$ modular form for $\Gamma$ and $f\vert_{\sigma}(z) = f(\sigma z)(cz + d)^{-k}$ where $\sigma = \begin{pmatrix} a & b\\c & d\end{pmatrix}$. Why is it true that $f$ is holomorphic at the cusps of $\Gamma$ if and only if $f\vert_{\sigma}$ is holomorphic at $\infty$ for all $\sigma \in SL_{2}(\mathbb{Z})$? (also couldn't this be weakened to say instead the $\sigma_{i} \in SL_{2}(\mathbb{Z})$ which the cusps $p_{i}$ to $\infty$?)

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The answers to both your questions follow immediately from the definition of cusp (together with the fact that $SL_2(\mathbb{Z})$ acts transitively on the cusps). Could you clarify, which part of the definition you are having trouble with? –  Alex B. Dec 26 '11 at 5:38
I'm having issues with proving that if $f$ is holomorphic at all cusps $p_{i}$, then $f|_{\sigma}$ is holomorphic at $\infty$ for all $\sigma \in SL_{2}(\mathbb{Z})$. I know that $f|_{\sigma}$ is holomorphic on the upper half plane and weakly modular for $\sigma^{-1}\Gamma\sigma$. –  Shayla Dec 26 '11 at 6:29
What is for you the definition of a cusp? –  Alex B. Dec 26 '11 at 6:45
My definition of cusp is a $\Gamma$-equivalence class of points in $\mathbb{Q}\cup\{\infty\}$. –  Shayla Dec 26 '11 at 6:48
... and what is your definition of "holomorphic at a cusp"? –  David Loeffler Dec 26 '11 at 12:14