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Consider the modular curve $\pi: X(N) \to X(1)$ where this map has Galois group $G = PSL_2(\mathbb Z/N\mathbb Z)$. In particular, $G$ acts on the singular cohomology $H^1(X(N),\mathbb Z)\otimes \mathbb C$ or in finite characteristic, on the etale cohomology group $H^1(X(N),\mathbb Z_\ell)\otimes_{\mathbb Z_\ell}\overline{\mathbb Q_\ell}$.

Do we know which irreducible representations of $G$ appear in the cohomology and with what multiplicities. Also, we can ask how the action of $G$ interacts with the Hecke operators, for instance. This seems to me to be very classical automorphic stuff but I have no knowledge about this area of math. Are there any friendly references?

Looking at the dimensions, I don't believe it is the regular representation.

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    $\begingroup$ This is classical, but I don't know a reference, One way to calculate this representation is by its character. You can use the fact that the trace of a nontrivial element is $2$ minus its number of fixed points of that element on $X(N)$. For any element that does not have order $2$, order $3$, or is unipotent there are no fixed points but for these special elements there are fixed points, so it's not quite the regular representation. $\endgroup$
    – Will Sawin
    Commented Aug 12, 2020 at 22:30
  • $\begingroup$ @DavidLoeffler Thank you! This is exactly the kind of thing I was looking for. The proof seems to be along the same lines as Sawin's suggestion. $\endgroup$
    – Asvin
    Commented Aug 13, 2020 at 10:53
  • $\begingroup$ Comment now reposted as an answer. $\endgroup$
    – David Loeffler
    Commented Aug 13, 2020 at 11:14
  • $\begingroup$ You can look at Kato, $p$-adic Hodge theory and values of zeta functions of modular forms, (4.9.3) and (4.9.4) for the question of how $G$ interacts with the Hecke operators. $\endgroup$
    – François Brunault
    Commented Aug 15, 2020 at 7:46

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Jared Weinstein's PhD thesis (http://math.bu.edu/people/jsweinst/jswthesis.pdf) is an excellent reference for this kind of thing. See section 3.4 in particular, where he computes the space $S_k(\Gamma(N), \mathbb{C})$ as a $\mathbb{C}[\mathrm{SL}_2(\mathbb{Z}/N)]$-module using an equivariant version of the Riemann--Roch formula.

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  • $\begingroup$ Just to make sure I understand things correctly: If I wanted to use a similar decomposition for etale cohomology instead, I would have to pass to an algebraic closure, use a comparison theorem with algebraic de rham which in turn by hodge theory is the sum of sheaf cohomology of $H^0(X,\omega)$ and it's dual. Since everything is functorial, this isomorphism also preserves the structure of the group action. Do I have that right? $\endgroup$
    – Asvin
    Commented Aug 16, 2020 at 2:14
  • $\begingroup$ Yes, that's right, assuming that by $\omega$ you mean the sheaf of differentials, and you take k=2 in Jared's formulae. $\endgroup$
    – David Loeffler
    Commented Aug 16, 2020 at 7:33
  • $\begingroup$ Yep, that's what I had in mind. Thanks a lot! $\endgroup$
    – Asvin
    Commented Aug 16, 2020 at 7:34

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