On the number of rational points of varieties and stacks

Manin-Peyre conjecture predicts the number of integral solutions of polynomial equations, i.e. the number of rational points on varieties. The predicted number is expressed by arithmetic and geometric invariants of the variety. Malle conjecture, predicts the number of field extensions of the field Q. Two predictions, despite the fact they are concerned with different questions, coming from different areas of number theory, appear very similar.

Stacks are geometrical objects, which are more general than varieties. They arise as solutions of classifying questions. Field extensions of Q are classified by rational points of certain stack. We try to motivate that there may exist a theory of Manin-Peyre conjecture for stacks which could have Malle conjecture for its consequence, and hence explain the above phenomenon.

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