Conjugations of arithmetic automorphic function fields

Abstract

In [1], Doi and Naganuma showed that the conjugation ofa Shimura curve is again a Shimura curve. The present paper deals with the generalization of their result.Consider in general a reductive algebraic group G defined over Q. Let G u be the semi-simple part of G. Assume that G~t modulo a maximal compact subgroup defines a bounded symmetric domain ~, and that a system of canonical models (in the sense of Shimura [11, 2.13]) for the quotients of ~ by the arithmetic subgroups Yx of G exists. Let {Vx, (gx, Jxw(U)} be such a system. Take an arbitrary automorphism z of the complex number field C, and conjugate all the Vx's and Jxw(U)'S by z. Then one expects that {V:~, C~x, Jxw(u) ~} with suitable ~x's forms a system of canonical models for some reductive group G1.In this paper we show that this is the case if G is the type of groups investigated by Shimura in [11]. The corresponding group G1 is defined explicitly in 1.2, and the precise statement of the result is given as Theorem 1.3. In Shimura's construction, for special Yx the model Vx is first realized as a s ubvariety of a moduli-variety Vo for some PEL-type f2. Consider V~ as embedded in V~. It is known that the conjugate variety V~is isomorphic to the moduli-variety V w of another PEL-type f2'. The relations between f2 and f2' are provided by Shimura's work [6]. One of our main task then is to prove that the isomorphism of V~ to V u, induces an isomorphism ofV~to Vxl for some arithmetic subgroup Fx~ of G1. This can be achieved by studying the isolated fixed points of G o and G1 o" We carry out these considerations in Sections 2 and 3.The same functorial property also holds for the models constructed by Miyake in [3]. In Section 4 we deal with this case briefly. We shall not give the proof, because the argument is similar to, and actually simpler than, the one presented in this paper.We assume the reader is familiar with Shimura's work [10] and [11], which will be quoted respectively as [A] and [C] hereafter.

Modular embeddings and automorphic Higgs bundles

“…This is a special case of a result of Milne-Shih [32], conjectured by Langlands [28]. For this case this result had been proved earlier in more classical language in [15] for Shimura curves and in [46] for quaternionic Shimura varieties of arbitrary dimension. The present author prefers the more abstract setup utilised here because it also leads to a useful description of the Galois actions on automorphic bundles, needed later on.…”

“…Proof. This is, in a slightly different language, the main result of [46]. It is stated in the language we use (but without proof and with a sign error) in [34,Remark 4.1(b)].…”

“…Note that the statement about the finite primes follows directly from (20); for the infinite primes some tedious but elementary calculations are needed. We refer to [46] for these, but it should be possible to replace them by the explicit cohomological calculations of [37].…”