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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$\begingroup$ That's the Eichler-Shimura relation for elliptic modular curves. I'm interested in the Shimura curve case. $\endgroup$– NicolásCommented Jun 7, 2011 at 15:47
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$\begingroup$ @unknown: Oops sorry $\endgroup$– C.S.Commented Jun 7, 2011 at 15:50
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$\begingroup$ Take a look at Zhang's paper in jstor.org/stable/2661372 $\endgroup$– A. PacettiCommented Jun 7, 2011 at 16:13
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1$\begingroup$ 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? $\endgroup$– Pete L. ClarkCommented Jun 7, 2011 at 17:16
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$\begingroup$ @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... $\endgroup$– Pete L. ClarkCommented Jun 7, 2011 at 18:30
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1 Answer
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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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2$\begingroup$ 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. $\endgroup$– OlivierCommented Jun 7, 2011 at 20:32
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$\begingroup$ 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). $\endgroup$ Commented Jun 7, 2011 at 20:54
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1$\begingroup$ 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. $\endgroup$– OlivierCommented Jun 8, 2011 at 5:25
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$\begingroup$ Does anyone happen to know a reference that explains the implication from Carayol's to Nekovar's formulation? Even in the case of $F=\mathbb{Q}$ this would be helpful. $\endgroup$– KonradCommented Aug 3, 2018 at 10:34