# Moduli interpretation of Hecke operators on Shimura curves

In his book on Automorphic Forms, Shimura gives (chapter 9) definitions of the the Hecke operators for Shimura curves.

One can give definitions of the Hecke and Atkin-Lehner operators in terms of the moduli interpretation of the curves $X_0(M)$. Shimura curves are of course solutions to a moduli problem. However, I can't find a similar description of the Hecke operators in this case.

$\textbf{Question:}$ Can somebody provide a reference giving a definition of the Hecke operators (in terms of moduli problem) for Shimura curves attached to orders in indefinite quaternion algebras.

(So we can, in particular, calculate the action in finite characteristic.)

• Are you sure Shimura curves have a good moduli interpretation? If I remember correctly, one of the reasons Carayol works so hard in his two paper opus is that only a non-canonical cover of the curve associated with an auxiliary choice of totally imaginary quadratic extension admits a moduli interpretation. – Keerthi Madapusi Pera Oct 13 '14 at 20:27
• Ah... I was thinking of curves over totally real fields strictly containing $\mathbb{Q}$. I think for the theory over $\mathbb{Q}$, you could look at Pete Clark's thesis here: math.uga.edu/~pete/thesis.pdf – Keerthi Madapusi Pera Oct 13 '14 at 22:06