Induced rings and the reduction of fields of Abelian modular functions

The main result of the present paper is that a ring obtained by reduction of Abelian modular functions is a ring of functions on a $p$-adic symplectic group over a perfect field $k$ of characteristic $p$, and the ring is induced by a homomorphism of a parabolic subgroup on $\operatorname{Aut}k$.