Multiple Mixing for Adele Groups and Rational Points

We prove an asymptotic formula for the number of rational points of bounded height on projective equivariant compactifications of H\G, where H is a connected simple algebraic group embedded diagonally into G := H. Introduction Let X ⊂ P be a smooth projective variety over a number field F . Fix a height function (1) H : P(F )→R>0 and consider the counting function N(X,T ) := {x ∈ X(F ) |H(x) ≤ T}. Manin’s conjecture [9] and its refinements by Batyrev–Manin [1], Peyre [17], and Batyrev–Tschinkel [3] predict precise asymptotic formulas for N(X◦, T ) as T→∞, where X◦ ⊂ X is an appropriate Zariski open subset of an algebraic variety with sufficiently positive anticanonical class. These formulas involve geometric invariants of X: • the Picard group Pic(X) of X; • the anticanonical class −KX ∈ Pic(X); • the cone of pseudo-effective divisors Λeff(X)R ⊂ Pic(X)R, and they depend on an adelic metrization L = (L, ‖ ·‖v) of the polarization L giving rise to the embedding X ⊂ P, i.e., on a choice of the height function in (1). Given these, one introduces the invariants: a(L), b(L), and c(L) so that the number of F -rational points on X◦ of L-height bounded by T is, conjecturally, given by (2) N(X◦,L, T ) = c(L) a(L)(b(L)− 1)! T a(L) log(T )b(L)−1(1 + o(1)), T→∞, see, e.g., [3] for precise definitions of the constants.