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I wanted to ask about the relationship between the congruence subgroups $\Gamma_1(N)\cap\Gamma_0(p)$ and $\Gamma_0(pN)$ where $p$ is a prime not dividing the integer $N$.

Does the space of cuspforms $S_k(\Gamma_0(pN))$ embbed in $S_k(\Gamma_0(p)\cap\Gamma_1(N))$? If so, as a direct summand?

Thank you!

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The first question is obvious : $\Gamma' \subset \Gamma \implies S_k(\Gamma) \subset S_k(\Gamma')$. The last point is more interesting.

  • For a $f \in S_k(\Gamma_1(N))$ and an integer $d$ coprime with $N$ let $\langle d \rangle f = f|_k \gamma$ where $\gamma \in \Gamma_0(N)$ whose bottom right entry $\equiv d\bmod N$.

    Invariance by $\Gamma_1(N)$ implies this operator $\langle d \rangle$ is well-defined (it depends only on $d \bmod N$, not on the particular choice of $\gamma$) and $ \langle d'd \rangle=\langle d' \rangle \langle d\rangle$.

  • For a Dirichlet character $\chi \bmod N$ let $$\pi_\chi f = \sum_{d \bmod N} \overline{\chi(d)} \langle d \rangle f$$ Thus $\langle d' \rangle \pi_\chi f= \chi(d') \pi_\chi f$ which means $\pi_\chi f \in S_k(\Gamma_0(N),\chi)$. Also $\pi_\chi \pi_\chi = \pi_\chi$ and

    $$\sum_{\chi \bmod N} \pi_\chi f = \phi(N) f$$ therefore $$S_k(\Gamma_1(N)) = \bigoplus_{\chi \bmod N} S_k( \Gamma_0(N),\chi)$$

  • For a $f \in S_k(\Gamma_1(N) \cap \Gamma_0(p))$ and a Dirichlet character $\chi \bmod N$ let $\tilde{\chi}(d) = \chi(d) 1_{\gcd(p,d)}$ the induced character $\bmod Np$. We obtain $$\sum_{\chi \bmod N} \pi_{\tilde{\chi}} f = \phi(N) f, \qquad \pi_{\tilde{\chi}} f \in S_k(\Gamma_0(Np),\tilde{\chi})$$

$$\implies S_k(\Gamma_1(N)\cap \Gamma_0(p)) = \bigoplus_{\chi \bmod N} S_k( \Gamma_0(Np),\tilde{\chi})$$

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  • $\begingroup$ great answer thank you! $\endgroup$
    – user1728
    Commented Sep 25, 2017 at 11:41

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