# Eichler-Shimura for Shimura curves

Hi,

What is the statement of the Eichler-Shimura relation for Shimura curves? And where can one find a proof?

Thanks

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That's the Eichler-Shimura relation for elliptic modular curves. I'm interested in the Shimura curve case. – Nicolás Jun 7 '11 at 15:47
@unknown: Oops sorry – S.C. Jun 7 '11 at 15:50
Take a look at Zhang's paper in jstor.org/stable/2661372 – A. Pacetti Jun 7 '11 at 16:13
If you choose $p$ prime to the discriminant of the quaternion algebra, I would expect it to be exactly the same as in the classical modular case. Is this not so? – Pete L. Clark Jun 7 '11 at 17:16
@Kevin: I guess you mean you have to stop to define the Hecke operator as a correspondence on the cusps, so we are quibbling about whether generalized enhanced elliptic curves are harder than (non-generalized) enchanced QM abelian surfaces? Anyway, no big deal either way... – Pete L. Clark Jun 7 '11 at 18:30

In the general case of Shimura curves over totally real number fields, a nice exposition of the Eichler-Shimura relation is given, for example, in $\S 1.14$ of this article of J. Nekovář, where you can find pointers to the relevant literature (in particular, a standard reference is Carayol's paper mentioned therein).