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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 $Y(U)$?

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

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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 $$Y(U)\times \mathbb Z/p^N\mathbb Z.$$

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