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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 $H^G$ of $H$ on the $Q$-vector space $V=H$: $u(x)=x u'$ $u(x)=x u'$$for all u\in H^ and x \in V=H and u \in G. Then$$E(u(x),u(y))=tr(\gamma x u'u y')=(u'u)tr(\gamma x y')$$y'),$$ which means that $$=Norm(u) E(u(x),u(y))=Norm(u) E(x,y).$$ This gives us the embedding $$H^* G \hookrightarrow GSp(V,E)\cong GSP(4.Q).$$ 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.

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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 group $H^$ on the $Q$-vector space $V=H$: $u(x)=x u'$ for all $u\in H^$ and $x \in V=H$. Then $$E(u(x),u(y))=tr(\gamma x u'u y')=(u'u)tr(\gamma x y')$$ $$=Norm(u) E(x,y).$$ This gives us the embedding $$H^* \hookrightarrow GSp(V,E)\cong GSP(4.Q).$$ The same construction over arbitrary commutative $Q$-algebras $R$ gives us the desired embedding of the corresponding $Q$-algebraic groups.