SATO-TATE IN THE HIGHER DIMENSIONAL CASE: ELABORATION OF 9.5.4 IN SERRE'S Nx(p) BOOK

摘要

In the very last paragraph of Serre's book Lectures on Nx(p), he writes "An interesting fact is that the Sato-Tate conjecture is sometimes easier to prove in the higher dimensional case (d > 1) than in the number field case, thanks to the information given by the geometric monodromy (as done by Deligne in characteristic p, cf. [De 80])." The purpose of this note is to spell out how this is done. In the higher dimensional case, one can bring to bear monodromy techniques. It turns out that a mild hypothesis "(H)" on the geometric monodromy is all that is needed; one gets a natural "Sato-Tate group" K in the sense of [Se-N_X(p), 8.2.2], in whose space of conjugacy classes the equidistribution takes place. Questions of modularity do not arise.