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.



The cohomology of modular curves as a module over the Galois group
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.


