Why a certain “Hecke polynomial” is equal to a certain “Galois polynomial”?

Let $f$ be a modular form of weight $k>0$ on $\Gamma_1(N)$ that is an eigenvector for all the Hecke operators $T_\ell$, for $\ell$ prime, and with Nebentype $\epsilon$. If $\ell$ is a prime not dividing $N$, set $\iota_\ell(f)(z):=f(\ell z)$. As it is well known, $f$ and $\iota_\ell(f)$ are distinct modular forms of weight $k$ on $\Gamma_1(N\ell)$ that span a two-dimensional space denoted by $V_f$. It is also well known that any Hecke operator $T_p$, for $p$ a prime other than $\ell$, acts on $V_f$ as a scalar, and that the characteristic polynomial of the linear action of $T_\ell$ (often denoted by $U_\ell$ in this context) on $V_f$ is $X^2-a_\ell X +\epsilon(\ell)\ell^{k-1}$, where $a_\ell$ is the $\ell$--th eigenvalue of our original $f\in M_k(\Gamma_1(N))$. A possible proof of this fact is a simple computation based on the explicit formulas for the effect of Hecke operators on $q$--expansions.

On the other hand, it is also well known that if $\lambda$ is a finite prime of the number field attached to $f$, and $\rho_\lambda$ is the two dimensional, $\lambda$--adic Galois representation of $Q$ attached to $f$, then the "Hecke polynomial" mentioned in the previous paragraph coincide with the characteristic ("Galois") polynomial of $\rho_\lambda$ at a Frobenius element at $\ell$, when $\lambda$ does not divide $\ell$.

Question: is there a conceptual proof of the equality between these two polynomials? Can you point a reference where this fact is explained? thanks.

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Eichler-Shimura. – Felipe Voloch Sep 20 '11 at 14:38
Thanks Felipe, can you please say something more? – Tommaso Centeleghe Sep 21 '11 at 8:50
The Eichler-Shimura congruence is a congruence between a Hecke operator and Frobenius viewed as correspondences on a modular curve. It implies what you are asking and that's essentially what it is. This should be explained in the standard places. Maybe if you ask a more specific question, you can get a more specific explanation. – Felipe Voloch Sep 24 '11 at 14:16
Dear Felipe, the source where I learnt of the Eichler-Shimura congruence is in one of latest chapters of Knapp's book "Elliptic curves". Even if I do not fully understand the technicalities, I believe that there it is shown that the correspondence on the modular curve X_0(N) (over Z[1/N]?) given by the \ell-th Hecke operator, for \ell not diving N, once it is base-changed to F_p, can be related to Frobenius and Verschiebung on elliptic curves by a direct computation performed over the ordinary locus. – Tommaso Centeleghe Sep 25 '11 at 17:47
Now, what this does have to do with the old space in level \ell N is not clear to me. I don't mean to contradict you but I think my question was very specific! :-) – Tommaso Centeleghe Sep 25 '11 at 17:50