# Description of a Shimura variety

Let $$(G, X)$$ be a Shimura datum and let $$U \subseteq G(\mathbb A_f)$$ be an open compact subgroup. By the general theory of Shimura varieties, we get a corresponding algebraic variety $$Y(U)$$ defined over a number field, say $$E$$. Now let $$\mathbb G_m$$ denote the multiplicative group and for $$N \in \mathbb N_{\geq 1}$$ and $$p$$ a prime, define $$V_N:=\{x \in \smash{\hat{\mathbb Z}}^{\times}: x \equiv 1 \bmod p^N \} \subseteq \mathbb G_m(\mathbb A_f)$$. What is the description of the Shimura variety associated to $$U \times V_N \subseteq (G \times \mathbb G_m)(\mathbb A_f)$$ in terms of $$Y(U)$$?

• How do you know if GxGm can be endowed with a Shimura datum (sorry if the question is naive..) ?? Aug 6, 2022 at 19:25

Unless I misunderstood, this seems straightforward. The Shimura variety you are asking for is $$(U\times V_N)\backslash (X\times G(\mathbb{A}_f)\times \mathbb{G}_m(\mathbb{A}_f))/(G(\mathbb Q)\times \mathbb Q^\times)= Y(U)\times (V_N\backslash\mathbb{A}_{f}^{\times}/\mathbb{Q}^{\times}).$$ So it is enough to determine the last factor. But as $$\mathbb{A}_{f}^{\times}=\hat{\mathbb Z}\times \mathbb Q^\times$$, then it simply $$\mathbb Z/p^N\mathbb Z$$. So the final answer is $$Y(U)\times \mathbb Z/p^N\mathbb Z.$$