On the distribution of points of bounded height on equivariant compactifications of vector groups. (English) Zbl 1067.11036

Chambert-Loir, Antoine; Tschinkel, Yuri

Invent. Math. 148, No. 2, 421-452 (2002).

Let \(F\) be a number field and \(X\) be an equivariant compactification of the group \({ G}^n_a\) over \(F\), projective and smooth. The authors of the article under review consider the (exponential) height function \(H\) associated to a class \(\lambda=\sum\lambda_{\alpha}D_{\alpha}\) contained in the cone of effective divisors (here \(D_{\alpha}\) are the components of the boundary divisor). They prove an asymptotic formula for the number of \(F\)-rational points in \({ G}^n_a\), with bounded height. Special situations were already investigated in [Compos. Math. 124, No. 1, 65–93 (1999; Zbl 0963.11033); J. Number Theory 85, No. 2, 172–188 (2000; Zbl 0963.11034)].
Let \(a_{\lambda}= \max(\rho_{\alpha}/\lambda_{\alpha})\) (where \(\rho_{\alpha}\) are the coefficients of the anticanonical line bundle) and \(b_{\lambda}\) be the cardinality of the set of \(\alpha\)’s where this maximum is reached. Then the number of \(F\)-rational points \(x\in { G}^n_a(F)\) with \(H(x)<B\) is asymptotically \(B^{a_{\lambda}}P_{\lambda}(\log B)\), with \(P\) a polynomial of degree \(b_{\lambda}-1\). The authors give an explicit formula for the leading coefficient of \(P\) and prove that it involves Peyre’s Tamagawa number when \(\lambda\) is the class of the anticanonical line bundle. They also evaluate the error term.
This formula follows from properties of the height zeta function, using a Tauberian theorem. The authors prove that the height zeta function \(Z_{\lambda}(s)\) converges absolutely and uniformly for \(Re(s)>a_{\lambda}\), has a meromorphic continuation to \(Re(s)>a_{\lambda}-\delta\) (for some \(\delta>0\)) with an unique pole at \(s=a_{\lambda}\) of order \(b_{\lambda}\), and has polynomial growth in vertical strips in this domain. In order to prove these properties, the authors extend the height function to the adelic space \({ G}^n_a({ A}_F)\) and represent \(Z_{\lambda}(s)\), by applying the additive Poisson formula, as a sum over the characters of \({ G}^n_a({ A}_F)/{ G}^n_a(F)\) of the Fourier transforms of \(H\). The Fourier transforms decompose as products of local transforms, which are computed at almost all finite places and are estimated for the remaining ones.

Reviewer: Razvan Litcanu (Iaşi)

MSC:

11G50	Heights
11G35	Varieties over global fields
14G05	Rational points
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

Keywords:

height; vector group; rational points of bounded height; height zeta function

Citations:

Zbl 0963.11033; Zbl 0963.11034