Shimura Varieties: the Adelic Point of View

S h → G R Ad → GL(g) Concretely, C× acts on gC via the characters z/z, 1, z/z. (2) ad(h(±i)) is a Cartan involution of G(R). (3) There is no Q-simple factor Gi of G such that S h → G R → G i,R is trivial. 1.2.2. Remark. Brian gave a slightly different defintion of a connected Shimura datum, using the continuous map u : S → G(R) obtained by restricting h(R) to the unit circle S ⊂ C× = S(R). This turns out to be equivalent, as he explained in §3 of his notes (or see [5, p. 50]). We now know that these are nice in several ways. First, X is a hermitian symmetric domain with a nice holomorphic action of G(R). This was more-or-less explained in the talks by Brandon, Martin, and Brian, but I’ll say something about it below to tie together what has been covered so far. Moreover, by the Baily–Borel theorem from Mike’s talk, for any torsion-free arithmetic subgroup Γ ⊂ G(R) the locally symmetric quotient ShΓ(G,X ) = Γ\X has the structure of a complex algebraic variety. Again, it’s worth being precise about exactly how the Baily–Borel theorem is being applied, so I will say something about this below. In this talk I shall explain another nice way to think about these quotients: when G is simply connected and Γ = G(Q)∩K ⊂ G(Af ) is a congruence subgroup (arising from a compact open subgroup K ⊂ G(Af )), then they can be described adelically as ShK(G,X ) := ShK∩G(Q)(G,X ) = G(Q)\(X ×G(Af ))/K. Furthermore, these varieties fit together into a nice inverse system with Sh◦(G,X+) := lim ←− K ShK(G,X ) = G(Q)\(X ×G(Af )).