Height zeta functions of equivariant compactifications of semi-direct products of algebraic groups

We apply the theory of height zeta functions to study the asymptotic distribution of rational points of bounded height on projective equivariant compactifications of semi-direct products. Introduction Let X be a smooth projective variety over a number field F and L a very ample line bundle on X. An adelic metrization L = (L, ‖ · ‖) on L induces a height function HL : X(F )→ R>0, let N(X◦,L,B) := #{x ∈ X◦(F ) |HL(x) ≤ B}, X◦ ⊂ X, be the associated counting function for a subvariety X◦. Manin’s program, initiated in [21] and significantly developed over the last 20 years, relates the asymptotic of the counting function N(X◦,L,B), as B→∞, for a suitable Zariski open X◦ ⊂ X, to global geometric invariants of the underlying variety X. By general principles of diophantine geometry, such a connection can be expected for varieties with sufficiently positive anticanonical line bundle −KX , e.g., for Fano varieties. Manin’s conjecture asserts that (0.1) N(X,−KX ,B) = c · B log(B)r−1, where r is the rank of the Picard group Pic(X) of X, at least over a finite extension of the ground field. The constant c admits a conceptual interpretation, its main ingredient is a Tamagawa-type number introduced by Peyre [25]. For recent surveys highlighting different aspects of this program, see, e.g., [37], [9], [7], [8]. Several approaches to this problem have evolved: • passage to (universal) torsors combined with lattice point counts; • variants of the circle method; • ergodic theory and mixing; • height zeta functions and spectral theory on adelic groups. 2000 Mathematics Subject Classification. Primary 11G35. 1 2 SHO TANIMOTO AND YURI TSCHINKEL The universal torsor approach has been particularly successful in the treatment of del Pezzo surfaces, especially the singular ones. This method works best over Q; applying it to surfaces over more general number fields often presents insurmountable difficulties, see, e.g., [14]. Here we will explain the basic principles of the method of height zeta functions of equivariant compactifications of linear algebraic groups and apply it to semi-direct products; this method is insensitive to the ground field. The spectral expansion of the height zeta function involves 1-dimensional as well as infinite-dimensional representations, see Section 3 for details on the spectral theory. We show that the main term appearing in the spectral analysis, namely, the term corresponding to 1-dimensional representations, matches precisely the predictions of Manin’s conjecture, i.e., has the form (0.1). The analogous result for the universal torsor approach can be found in [26] and for the circle method applied to universal torsors in [27]. Furthermore, using the tools developed in Section 3, we provide new examples of rational surfaces satisfying Manin’s conjecture. Acknowledgments. We are grateful to the referee and to A. Chambert-Loir for useful suggestions which helped us improve the exposition. The second author was partially supported by NSF grants DMS-0739380 and 0901777. 1. Geometry In this section, we collect some general geometric facts concerning equivariant compactifications of solvable linear algebraic groups. Here we work over an algebraically closed field of characteristic 0. Let G be a connected linear algebraic group. In dimension 1, the only examples are the additive group Ga and the multiplicative group Gm. Let X∗(G) := Hom(G,Gm) be the group of algebraic characters of G. For any connected linear algebraic group G, this is a torsion-free Z-module of finite rank (see [36, Lemma 4]). Let X be a projective equivariant compactification of G. If X is normal, then it follows from Hartogs’ theorem that the boundary D := X \G, is a Weil divisor. Moreover, after applying equivariant resolution of singularities, if necessary, we may assume that X is smooth and that the boundary D = ∪ιDι, is a divisor with normal crossings. Here Dι are irreducible components of D. Let Pic(X) be the group of equivalence classes of G-linearized line bundles on X. Generally, we will identify divisors, associated line bundles, and their classes in Pic(X), resp. Pic(X). Proposition 1.1. Let X be a smooth and proper equivariant compactification of a connected solvable linear algebraic group G. Then, (1) we have an exact sequence 0→ X∗(G)→ Pic(X)→ Pic(X)→ 0, (2) Pic(X) = ⊕ι∈IZDι, and HEIGHT ZETA FUNCTIONS 3 (3) the closed cone of pseudo-effective divisors of X is spanned by the boundary components: Λeff(X) = ∑