Monodromy of Hecke-invariant subvarieties

Abstract

The main result of this note is that for any primes ` 6= p, the `-adic monodromy group of the Zariski closure H(x0) of the `-adic Hecke orbit of a non-supersingular point x0 in Ag over Fp is equal to the full symplectic group; see Prop. 4.1. Here Ag denotes the moduli space of g-dimensional principally polarized abelian varieties. The proof consists mostly of group theory. It does use some nontrivial information from algebraic geometry, namely Grothendieck’s semisimplicity theorem for pure Q`-sheaves and the Riemann hypothesis for abelian varieties over finite fields.