Suppose we have a PEL type $(H,\phi ,\*;T,O,V)$ where H is a rational nonsplit quaternion algebra, $\phi$is an embedding of Q-algebra $\phi : H-->M(2,R)$, and \* is a positive anti involution of H; O is the maximal order of H , and V level structure . Associate these datum a shimura curve parametric fake elliptic curves in a standar way.
My question is :what is the group for the shimura datum ,is it the group H\*, the invertible element of H ? And since this family parametirc abelian two folds , what is the map from this group to $\mathrm{GSP}(4,\mathbb{Q})$?



Shimura datum of family of fake elliptic curves
Actually, fake elliptic curves are discussed in Chapter 9 of Lang's Introduction to algebraic and abelian functions. 


In order to describe the map to $GSP(4,Q)$, recall that there is the standard antiinvolution $x\to x'$ on the quaternion $Q$-algebra $H$ such that both $tr(x)=x+x'$ and $Norm(x)=xx'=x'x$ are rational numbers for all $x\in H$. Every positive antiinvolution of $H$ is of the form $x \to \gamma^{-1}x'\gamma$ where $\gamma$ is a fixed nonzero (invertible) element of $H$ such that $\gamma^2$ is a negative rational number and therefore $\gamma'=-\gamma$. This gives rise to the alternating nondegenerate $Q$-bilinear form
$$E: H \times H \to Q, x,y \mapsto tr(\gamma x y').$$
Now let us consider the following faithful action of the multiplicative group $G$ of $H$ on the $Q$-vector space $V=H$:
$$u(x)=x u'$$ for $x \in V=H$ and $u \in G$. Then


$$E(u(x),u(y))=tr(\gamma x u'u y')=(u'u)tr(\gamma x y'),$$


which means that
$$E(u(x),u(y))=Norm(u) E(x,y).$$


This gives us the embedding $G \to GSP(V,E)\cong GSP(4.Q)$.


The same construction over arbitrary commutative $Q$-algebras $R$ gives us the desired embedding of the corresponding $Q$-algebraic groups.


