Conjugates of Rational Equivariant Holomorphic Maps of Symmetric Domains

Abstract

Let \(\tau: {\cal D} \rightarrow{\cal D}^\prime\) be an equivariant holomorphic map of symmetric domains associated to a homomorphism \({\bf\rho}: {\Bbb G} \rightarrow{\Bbb G}^\prime\) of semisimple algebraic groups defined over \({\Bbb Q}\). If \(\Gamma\subset {\Bbb G} ({\Bbb Q})\) and \(\Gamma^\prime \subset {\Bbb G}^\prime ({\Bbb Q})\) are torsion-free arithmetic subgroups with \({\bf\rho} (\Gamma) \subset \Gamma^\prime\), the map τ induces a morphism φ: \(\Gamma\backslash {\cal D} \rightarrow\Gamma^\prime \backslash {\cal D}^\prime\) of arithmetic varieties and the rationality of τ is defined by using symmetries on \({\cal D}\) and \({\cal D}^\prime\) as well as the commensurability groups of Γ and Γ′. An element \(\sigma \in {\rm Aut} ({\Bbb C})\) determines a conjugate equivariant holomorphic map \(\tau^\sigma: {\cal D}^\sigma \rightarrow{\cal D}^{\prime\sigma}\) of τ which induces the conjugate morphism \(\phi^\sigma: (\Gamma\backslash {\cal D})^\sigma \rightarrow(\Gamma^\prime \backslash {\cal D}^\prime)^\sigma\) of φ. We prove that τσ is rational if τ is rational.