Height zeta functions of twisted products
The authors announce the following result, the complete proof of which is deferred to a future paper (to appear in Selecta Math.), but the basic arguments are sketched in the paper under review. Let \(X\) be a smooth projective toric variety over a number field \(F\) which is an equivariant compactification of the split algebraic torus \(T\). Let \(G\) be a semisimple split linear algebraic group over \(F\) with \(F\)-rational parabolic subgroup \(P\), and fix a morphism \(P \to T\). Then one can define twisted products \(Y = X \times^P G\) and \(Y^\circ = T \times^P G \subset Y\) (which are locally trivial bundles over \(P \backslash G\) with fibers \(X\) and \(T\), respectively). Let \({\mathcal L}\) be a metrized line bundle on \(Y\) (with a specific metrization) such that its class \([L]\) in \(\text{Pic}(Y) \otimes{\mathbb R}\) is contained in the interior of the cone of effective divisors. Then \({\mathcal L}\) defines a height function \(H_{\mathcal L} : Y(F) \to{\mathbb R}_{>0}\), and one can define the \textit{height zeta function} by \(Z(Y^\circ, {\mathcal L}, s) = \sum_{x \in Y^\circ(F)} H_{\mathcal L}(x)^{-s}\). Under the assumptions made, there are numbers \(a \in{\mathbb Q}_{> 0}\), \(b \in{\mathbb Z}_{> 0}\), \(\Theta \neq 0\), \(\delta > 0\) and a function \(h(s)\) that is holomorphic for \(\text{Re}(s) \geq a - \delta\), such that \[ Z(Y^\circ, {\mathcal L}, s) = \frac{\Theta}{(s - a)^b} + \frac{h(s)}{(s - a)^{b-1}} \] for \(\text{Re}(s) \geq a - \delta\). The numbers \(a\) and \(b\) only depend on \([L]\). By a standard Tauberian theorem, this implies an asymptotic formula for the number of rational points on \(Y^\circ\) of bounded height, proving a conjecture of \textit{Batyrev} and \textit{Yu. I. Manin} [Math. Ann. 286, 27-43 (1990; Zbl 0679.14008] for the class of varieties under consideration.
scientific article; zbMATH DE number 1053508