Reflex fields of Shimura varieties 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?
 A: 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.
A: 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.
