Equidistribution for Tannakian monodromy groups

We prove that a perverse sheaf on a connected commutatitve algebraic group over a finite is generically unramified. This implies an equidistribution theorem for Tannakian monodromy groups in previously unavailable generality. We also prove a stratification theorem for exponential sums in families indexed by a scheme and the characters of a connected commutative algebraic group. Our method is based on Tannakian categories introduced by Gabber and Loeser. This method naturally yields fiber functors. We also prove vanishing theorems over a connected commutative algebraic group, classify the negligible sheaves, and prove relative weak propagation theorems for tori.