# Carayol's "ramified Eichler-Shimura relation" and its applications

In his paper "Sur la mauvaise reduction des courbes de Shimura" from '86 H. Carayol shows the following congruence relation:

Let $$M$$ be the tower of Shimura curves over a totally real $$F$$, associated to a quaternion algebra over $$F$$ that is split at one infinite place.

Fix a prime $$\mathfrak{p}$$ and consider the base change of $$M$$ to $$F_\mathfrak{p}$$. It carries a $$p$$-divisible group $$E_\infty$$ and using this, he constructs integral models $$\mathbb{M}$$ and $$\mathbb{E}_\infty$$ over $$\mathcal{O}_\mathfrak{p}$$.

The congruence relation (§ 10.3) is an assertion on the geometric special fiber of this model: It carries actions by $$\rm GL_2(F_\mathfrak{p})$$ and the unramified quotient of the Weil group $$W(F_\mathfrak{p}^{\rm ab}/F_\mathfrak{p})$$. I think these actions commute, since all geometric objects occuring, including all morphisms are actually defined over $$\mathcal{O}_\mathfrak{p}$$.

Each point $$A$$ in $$\mathbb{P}^1(F_\mathfrak{p})$$ determines a closed subscheme and the matrices $$X:=(\begin{smallmatrix} u & * \\ 0 & 1\end{smallmatrix})$$ stabilize the subscheme corresponding to $$[1,0]$$.

The congruence relation is: Each point in this subscheme is stabilized by $$(X,\rm{Frob}^{val(u)})$$ and this induces on $$\mathbb{E}_\infty|_x$$ the $$val(u)$$-power of absolute Frobenius.

Now to the application and my question: In § 5.2 of his paper "Sur les representations $$\ell$$-adic associées aux formes modulaires de Hilbert" (also from '86) Carayol applies the congruence relation to show that local-global compatibility holds in the Langlands correspondence for Shimura curves (and hence Hilbert modular forms of appropriate type).

He constructs $$\lambda$$-adic sheafs on $$M_{et}$$ and $$\mathbb{M}_{et}$$ much like $$E_\infty$$ and $$\mathbb{E}_\infty$$ (but instead of the standard representation, some weight enters).

The etale cohomology of the geometric special fiber at some prime $$\mathfrak{p}$$ then again carries (by functoriality of cohomology) an $$\rm GL_2(F_\mathfrak{p}) \times W(F_\mathfrak{p}^{\rm ab}/F_\mathfrak{p})$$-action.

Now to my question: Carayol mentions that by the previous result the diagonal action (as above and suitably restricted to the $$[1,0]$$-part of cohomology) on cohomology must be trivial. Why is this?

You might say this is easy: that absolute Frobenius acts trivially on cohomology. Yes, I understand the result for the etale cohomology of $$\mathbb{E}_\infty$$, or even nontrivial-weight-twists of it.

But the crucial point in the first paper is that the coefficients are a $$\mathfrak{p}$$-adic sheaf, while in the second paper $$\lambda$$ and $$\mathfrak{p}$$ very much are of different residue characteristic (T. Saito did the p=l case 15 years later).

I hope I could make the point of my confusion clear and apologize if this is a notational triviality.