We discuss existence and abundance of Galois-generic points for adelic representations
attached to Shimura varieties. First, we show that, for Shimura varieties of abelian type,
-Galois-generic
points are Galois-generic; in particular, adelic representations attached to such Shimura
varieties admit (“lots of”) closed Galois-generic points. Next, we investigate further the
distribution of Galois-generic points and show the André–Pink conjecture for them:
if
is a connected Shimura variety associated to a
-simple reductive
group, then every infinite subset of the generalized Hecke orbit of a Galois-generic point is
Zariski-dense in .
Our proof follows the approach of Pink for Siegel Shimura varieties. Our main
contribution consists in showing that there are only finitely many Hecke operators of
bounded degree on (adelic and connected) Shimura varieties. Compared with other
approaches of this result, our proof, which relies on Bruhat–Tits theory, is effective
and works for arbitrary Shimura varieties.
Keywords
Shimura varieties, Hecke orbits, Adelic representations of
étale fundamental group, Galois generic points