Counting points and Galois extensions

I will talk about the relation between counting points of an algebraic variety and counting Galois extensions. Quotient varieties and algebraic stacks play the role of bridge between these two counting problems. Working over a global field, the relation leads to relating/unifying the Batyrev-Manin conjecture on distribution of rational points and Malle’s conjecture on distribution of Galois extensions. If we work over a local field instead, then we get a generalization of the McKay correspondence to positive/mixed characteristics, which has applications to the study of quotient singularities and to mass formulas for extensions of local fields.