American Journal of Mathematics, 1981, Vol. 103, No. 5, pp. 911--935.

Comments

We study the automorphisms of Shimura varieties, and we deduce the existence of canonical models in the sense of Shimura from knowing the existence of canonical models in the sense of Deligne. Contrary to some expert opinion at the time, this was a serious exercise involving results not known to the afore-mentioned experts. Briefly, the results Deligne involve automorphisms of the Shimura variety whereas Shimura's conjecture is in terms of automorphisms of the function fields; to relate them, we needed theorems of K. Miyake and T. Miyake.

In the article, we consider only Shimura varieties defined by classical groups. This was not much of a restriction at the time, because canonical models were only known to exist in that case. By allowing us to describe the reductive group in terms of semisimple algebras, it enabled us to give more elementary proofs. In fact, by using the more abstract group-theoretic results of Harish-Chandra and Borel, one can remove this restriction without difficulty.

I plan (2019) to rewrite the article so that it applies to all Shimura varieties. I hope this will help those familiar with Deligne's point-of-view to understand Shimura's point-of-view.

Erratum

p912 $E(h)$ should be $E(z)$.