Height zeta functions
Let \(V\) be a projective algebraic variety defined over a number field \(k\) and let \(H: V(k)\to\mathbb R_+\) be a height function. Given an open subset \(U\) of \(V\), let \[ N_{U,H}(B):= \#\{\vec x\mid\vec x\in U(k),\,H(\vec x)\leq B\}. \] Assuming that the set of rational points \(V(k)\) is Zariski dense in \(V\) and that \(N_{U,H}(B)> 0\) for sufficiently large \(B\), one studies the asymptotic behaviour of the counting function \(N_{U,H}(B)\) as \(B\to\infty\). This question is intimately related to the analytic properties of the ``height zeta function'' \[ s\mapsto Z(s),\quad s\in\mathbb C,\quad Z(s):= \sum_{\vec x\in U(k)} H)\vec x)^{-s}. \] An asymptotic formula of the shape \[ N_{U,H}(B)= BP(\log B)+ O(B^\delta),\;P(t)\in\mathbb R[t],\;\deg P= r,\;0\leq\delta< 1, \] implies that the Dirichlet series, defining \(Z\), converges absolutely for \(\text{Re}\,s> 1\) and the function \(s\mapsto Z(s)\) can be analytically continued to the half-plane \(\mathbb C_\delta:= \{s\mid s\in\mathbb C,\text{Re}\,s> \delta\}\), \(\delta< 1\), as a meromorphic function with the only pole of order \(r+ 1\) at \(s= 1\). In a series of joint works of R. de la Bretèche with Browning, Derenthal, and Swinnerton-Dyer, the authors meromorphically continue the height zeta functions to the half-plane \(\mathbb C_\delta\), \(0< \delta< 1\), and prove asymptotic formulae as above for a few algebraic varieties defined over \(\mathbb Q\). The paper under review surveys those works. The author cites, for instance, among others the following results from the work of \textit{R. de la Bretèche} and Sir \textit{P. Swinnerton-Dyer} [Bull. Soc. Math. Fr. 135, No. 1, 65--92 (2007; Zbl 1207.11068)]. Let \(V\) be a projective hypersurface defined by the equation \(x^3_0= x_1x_2x_3\), \(\vec x:= (x_0,x_1,x_2,x_3)\), let the open subset \(U\) of \(V\) be defined by the condition \(x_0\in 0\), and let \[ H(\vec a)= \max_{0\leq i\leq 3}|a_i| \] for \(\vec a\in\mathbb Z^4\) with hcf. \((a_0,\dots, a_3)= 1\). Then the function \(s\mapsto Z(s)\) can be meromorphically continued to the half-plane \(\mathbb C_{3/4}\) and the line \(\{s\mid s\in\mathbb C,\,\text{Re\,}s= 3/4\}\) is the natural boundary for its analytic continuation. Moreover, if the Riemann hypothesis is true, then the following asymptotic formula holds: \[ N_{U,H}(B)= BP(\log B)+ aB^{9/11}+ O(B^\delta),\;a> 0,\;P(t)\in\mathbb R[t], \] for any \(\delta> 13/16\).
scientific article; zbMATH DE number 5620669
scientific article