# Definition of Hecke operators for Shimura curve

I am following Wiles' paper On class groups of imaginary quadratic fields, and it's my first time learning about Shimura curves. There is part of the setup that I don't understand, concerning the definition of the Hecke correspondences on Shimura curves over $$\mathbf{Q}$$.

Let $$B$$ be an indefinite quaternion algebra over $$\mathbf{Q}$$, let $$G/\mathbf{Q}$$ be the corresponding algebraic group, and let $$K \subset G(\mathbf{A}_f)$$ be an open compact subgroup (the level). Then Wiles defines the Shimura curve corresponding to this data, $$M_K = G(\mathbf{Q}) \backslash G(\mathbf{A}_f) \times \mathbf{H}^{\pm} / K,$$ where $$\mathbf{H}^{\pm} = \mathbf{C} - \mathbf{R}$$. In fact, Wiles further restricts to the case where $$K$$ is given by the product of local components $$K_p$$ where $$K_p$$ is the unit group of a maximal order in $$B \otimes \mathbf{Q}_p$$ for finite primes $$p$$ where $$B$$ is ramified, and otherwise (when $$B$$ is split at $$p$$) is so that $$K_p \subset B \otimes \mathbf{Q}_p = M_2(\mathbf{Q}_p)$$ has determinants surjective onto $$\mathbf{Z}_p^\times$$. So far so good.

I am a little confused about the definition of the Hecke correspondences on $$M_K$$ which are given next. It is as follows:

Let $$\mathcal{O}_B$$ be a maximal order in $$B$$, and $$\widehat{\mathcal{O}}_B = \mathcal{O}_B \otimes \widehat{\mathbf{Z}}$$. Let $$m$$ be a positive integer such that for every $$p | m$$, $$K_p$$ is a maximal order. We allow the case where $$B$$ is ramified at $$p$$. Let $$G_m$$ be the set of elements $$g \in \widehat{O}_B$$ which have component $$1$$ at primes not dividing $$m$$ and such that $$\mathrm{det}(g)$$ generates the ideal $$m\mathbf{Z}_p$$ at each prime $$p | m$$. In particular, $$G_1$$ is a subgroup of $$K$$, and $$G_m$$ is a union of cosets of $$G_1$$ in $$\widehat{\mathcal{O}}_B$$. We define an correspondence $$T_m$$ on $$M_K$$ by the formula $$T_m(y) = \sum_{\gamma \in G_m \setminus G_1}[(g \gamma, x)]$$ where $$y \in M_K$$ is represented in $$G(\mathbf{A}_f) \times \mathbf{H}^{\pm}$$ by $$(g, x)$$.

I am confused for the following reasons. First, isn't $$G_1$$ just $$1$$, since its elements have component $$1$$ at all primes? Also, when $$m = 1$$, regardless of whether I am right or wrong about what $$G_1$$ is, the formal sum here is empty. Shouldn't I expect $$T_1$$ to be the identity instead?

• They meant unit component instead of component $1$ ? And $G_m \setminus G_1$ is the quotient of a set by the group action of $G_1$, not a setminus Dec 17, 2020 at 7:51
• @reuns: thanks ! So the \ is a typo for a forward slash, then? Dec 17, 2020 at 16:46