My understanding is that the Eichler-Shimura relation expresses the Hecke operator $T\_p$ in terms of the geometric Frobenius map. Specifically, $T\_p = Frob + Ver$ for Frobenius map $Frob$ and it's transpose $Ver$. 


However, Wikipedia states that "the Eichler–Shimura congruence relation expresses the local L-function of a modular curve at a prime p in terms of the eigenvalues of Hecke operators".


I have no idea how that statement comes from the above definition of the congruence. How does such a decomposition of Hecke operators have anything to with its eigenvalues or L-functions of a modular curve?


EDIT: Will's amazing answer gives the connection between L-functions and eigenvalues, but how is the local L-function of a modular curve restated explicitly using Eichler-Shimura?



How is the Eichler-Shimura congruence related to L-functions?
Recall that the local $L$-function of the modular curve $X$ is $$\frac{1}{\det \left(1 - p^{-s} \operatorname{Frob}\_p, H^1(X, \mathbb Q\_\ell)\right)}.$$


The denominator is simply a variant of the characteristic polynomial of Frobenius, o it is sufficient to relate the eigenvalues of Frobenius to the Hecke eigenvalues.


Now because $FV = p$, $F$ and $V$ commute, and so for each $\lambda$ an eigenvalue of $F$, $\lambda+ p/\lambda$ is an eigenvalue of $F+V$.


Using this, one can give formulas for the characteristic polynomial of the Hecke operator in terms of the characteristic polynomial of Frobenius, or vice versa.


