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I am currently learning the theory of Shimura varieties. Out of curiosity, is it known which number fields can occur as reflex fields? More precisely, can one find, for any number field, a positive dimensional Shimura variety which has this field as its reflex field?

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  • $\begingroup$ My guess is that they're all totally real or CM? My logic is that even though I don't recall the precise definition, all the ones I've seen have been totally real or CM :-) $\endgroup$ – eric Dec 16 '15 at 19:09
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The answer depends on your definition of a Shimura pair $(G,X)$.

Look in Section 2.1 of Deligne's paper. If you assume only axioms (2.1.1.1), (2.1.1.2) and (2.1.1.3), then any number field $F$ can occur as the reflex field $E(G,X)$ with $X$ of positive dimension. Indeed, take $G_1=R_{F/\mathbf{Q}}\mathbb{G}_{m,F}$, then for suitable $h_1\colon\mathbf{S}\to G_{1,\mathbf{R}}$ we have $E(G_1,h_1)=F$. We take $X_1=\{h_1\}$, then $\mathrm{dim}_\mathbf{C}(X_1)=0$. We take $(G_2,X_2)=(\mathrm{GL}_{2,\mathbf{Q}},X_2)$ (the standard Shimura pair for $\mathrm{GL}_{2}$), then $E(G_2,X_2)=\mathbf{Q}$, $\mathrm{dim}(X_2)=1$. Set $G=G_1\times_\mathbf{Q} G_2$, $X=X_1\times X_2$. Then $\mathrm{dim}_\mathbf{C}(X)=1$, $E(G,X)=E(G_1,X_1)=F$.

However, if you assume also axioms (2.1.1.4) and (2.1.1.5), then $E(G,X)$ must be either a totally real field or a CM-field. In order to show this, it suffices to consider the case when $G$ is $\mathbf{Q}$-simple adjoint and the case when $G$ is a torus. For the $\mathbf{Q}$-simple adjoint case see Section 2.3.4 and Proposition 2.3.6 in Deligne's paper. In the toric case, axioms (2.1.1.4) and (2.1.1.5) imply that the torus $G$ is isogenous to the product of $\mathbb{G}_{m,\mathbf{Q}}$ and a $\mathbf{Q}$-torus which is compact over $\mathbf{R}$, hence it splits over a CM-field, and the assertion follows.

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Consider the Shimura variety defined by a datum $(G,X)$, and let $T$ denote the quotient of $G$ by its derived group, so $T$ is a torus over $\mathbb{Q}$. If $T$ splits over a CM-field then the reflex field is either totally real or CM. This takes care of most "naturally occurring" Shimura varieties. However, according to Deligne's definition, you get a Shimura variety from any torus over $\mathbb{Q}$ and cocharacter, and then the reflex field is the field of definition of the cocharacter, which can probably be anything.

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