Isogenies, Connectedness, and Lie-Irreducibility

This chapter takes up the proofs of Theorems 8.1 and 8.2. For each prime to p integer n, we have the n'th power homomorphism [n] : G → G. Formation of the direct image is an exact functor from Perv to itself, which maps Neg to itself, in Ƿ to itself, and which (because a homomorphism) is compatible with middle convolution. So for a given object N in Garith, [n]* allows us to view 〈N〉arith as a Tannakian subcategory of 〈[n]* N〉arith, and 〈N〉geom as a Tannakian subcategory of 〈[n]* N〉geom.