# 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

## 1 Answer

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).

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Though one should point out that in fact the exposition in 1.14 of this article is faulty: the expression given is valid for the arithmetic Frobenius, but the geometric Frobenius is used. Carayol's Sur la mauvaise réduction des courbes de Shimura section 10.3 is impeccable. –  Olivier Jun 7 '11 at 20:32
Thanks a lot for the comment, Olivier! I admit that, having usually referred to Carayol's paper, I had overlooked this issue in Nekovář's article (which - with this point of caution in mind - offers a somewhat more friendly exposition of the Eichler-Shimura stuff, in my opinion). –  Stefano V. Jun 7 '11 at 20:54
And now Christophe Cornut tells me that one should double check Carayol's article too: apparently, he overlooked the sign error in Deligne's Corvallis article. So maybe the best reference is Cornut Vatsal CM Points and Quaternion Algebras section 3.2.3. –  Olivier Jun 8 '11 at 5:25