Upper Motives of Products of Projective Linear Groups

This paper is devoted to the classification of the upper motives of all the products of projective linear groups with coefficients in a finite field.We first give a complete classification of the upper motives of all the projective linear groups and derive from it the motivic dichotomy of PGL₁.We then provide several classification results as well as counterexamples for arbitrary products of projective linear groups, showing that the situation is less rigid. The proofs involve a neat study of rational maps between generalized Severi-Brauer varieties which is certainly of independent interest.