Geodesic
Induced rings and the reduction of fields of Abelian modular functions
Izvestiya. Mathematics , Tome 4 (1970) no. 3, pp. 536-550

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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 $\operatorname{Aut}k$. 
@article{IM2_1970_4_3_a4,
     author = {I. I. Pyatetskii-Shapiro},
     title = {Induced rings and the reduction of fields of {Abelian} modular functions},
     journal = {Izvestiya. Mathematics },
     pages = {536--550},
     publisher = {mathdoc},
     volume = {4},
     number = {3},
     year = {1970},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1970_4_3_a4/}
}
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I. I. Pyatetskii-Shapiro. Induced rings and the reduction of fields of Abelian modular functions. Izvestiya. Mathematics , Tome 4 (1970) no. 3, pp. 536-550. http://geodesic.mathdoc.fr/item/IM2_1970_4_3_a4/