Let $G$ be a connected reductive algebraic group over $\mathbf{Q}$. I've seen two slightly different definitions in the literature of the Shimura variety of level $U$, for $U \subseteq G(\mathbf{A}_{\mathrm{f}})$ an open compact subgroup.

Milne's "Introduction to Shimura varieties", following Deligne's "Travaux de Shimura", defines $$Sh_G(U) = G(\mathbf{Q})\, \backslash\, G(\mathbf{A})\, /\, U K_\infty$$ where $K_\infty \subseteq G(\mathbf{R})$ is the centraliser of the cocharacter $h: \mathbf{S} \to G_{\mathbf{R}}$ defining the Shimura datum. Deligne shows in Prop 2.6 of "Travaux de Shimura" that $$K_\infty = (\text{max compact subgroup of $G'(\mathbf{R})^\circ$}) \cdot Z_G(\mathbf{R}),$$ where $G' = $derived subgroup, and "$\circ$" denotes "identity component".

However, I've also seen a definition using a slightly different recipe: $$\widetilde{Sh}_G(U) = G(\mathbf{Q})\, \backslash\, G(\mathbf{A})\, /\, U \widetilde{K}_\infty$$ where $$\widetilde{K}_\infty = \text{(max compact subgroup of $G(\mathbf{R})^\circ)$} \times Z_G(\mathbf{R})^\circ.$$ E.g. this definition is the one used in Emerton's "On the interpolation of systems of Hecke eigenvalues" (Inventiones 2006).

These definitions really seem to be different in general. If $G$ is semisimple and adjoint (so $Z_G = \{1\}$), they are the same. If $G$ is a torus, then they differ by the component group of $G(\mathbf{R})$; in fact Milne's lecture notes mention something equivalent to $\widetilde{Sh}_G(U)$ in the setting of tori. [With Milne's definitions, a "0-dimensional Shimura variety" is not the same as a Shimura variety which happens to be 0-dimensional!]

For general $G$ it's not obvious to me if there's even a map between the two. EDIT: If I'm not mistaken, we can conjugate $K_\infty$ and $\tilde K_\infty$ so they have the same Lie algebra; $\tilde K_\infty$ is automatically connected, although $K_\infty$ may not be, so $\tilde K_\infty$ is always contained in $K_\infty$. Thus $\widetilde{Sh}_G(U)$ is a finite covering of $Sh_G(U)$.

Is it true that the compatible system of varieties $\widetilde{Sh}_G(U)$ has a model over the reflex field?

(I'm having trouble coming up with "interesting" examples where $K_\infty \ne \tilde{K}_\infty$ -- the only examples I can think of are tori, where the existence of models is trivial anyway. This question is about trying to find interesting examples.)

  • 2
    $\begingroup$ In the definition of Shimura variety, $X$ should be replaced by a set $Y$ with a transitive action of $G(\mathbb{R})$ which is a finite covering of the usual $X$. Pink pointed out that this becomes necessary when considering the boundaries of Shimura vareties (and it would justify Milne's terminology). Probably the whole of the theory will continue to work in this more general context. $\endgroup$ – anon Jun 13 at 12:48
  • $\begingroup$ @anon That sounds very plausible, + would give a proper context for both the "usual" theory and Milne's treatment of tori; but do you know where this is written down? $\endgroup$ – David Loeffler Jun 13 at 16:53
  • $\begingroup$ Pink discusses this in his thesis, page 26 et seq. (Arithmetical compactification of mixed Shimura varieties. Bonn 1989/1990). Apart from his thesis, I don't know anywhere this is systematically studied. $\endgroup$ – anon Jun 13 at 18:13

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