Counting rational points on smooth cyclic covers (Q423619)

The paper addresses Serre's conjecture about the number of rational points of bounded height on a finite cover of the projective space \({\mathbb P}^{n-1}\). Given a finite cover \(\phi\,:\,X \to {\mathbb P}^{n-1}\) over \({\mathbb Q}\), one introduces the counting function NEWLINE\[NEWLINE N_B(\phi) = \#\{P\in X({\mathbb Q}) : H(\phi(P)) \leq B\}. NEWLINE\]NEWLINE The conjecture is that \(N_B(\phi) \ll B^{n-1} (\log B)^c\) for some \(c\) and for covers of degree \(r\geq 2\).NEWLINENEWLINEThe authors investigate the number of integer solutions with \(|x_i|\leq B\) to \(y^r = F(x_1,\dots,x_n)\), where \(F\) is an irreducible form with integer coefficients of degree \(mr\), with \(r\geq 2\) and \(m\geq 1\), and such that the projective hypersurface defined by \(F(x_1,\dots,x_n)=0\) is smooth. An upper bound for the number of such points of course gives an upper bound for the number of \textit{cyclic} covers of \({\mathbb P}^{n-1}\).NEWLINENEWLINEIn fact, the authors can get rid of the condition that the polynomial \(F\) shoud be a form, allowing \(F\) to be a general polynomial of degree \(d\geq 3\), subject to the only condition that its leading form is nonsingular.NEWLINENEWLINETheir main result is stated in terms of a quantity called \(N_{w,B}(F)\), where \(w : {\mathbb R}^n \to {\mathbb R}\) is a suitable non-negative weight, and which relates to the original counting function by the inequality \(N_B(\phi)\leq N_{w,2B}(F)\). The statement is the following: for every \(r\geq 2\)NEWLINENEWLINENEWLINE\[NEWLINE N_{w,B}(F) \ll \begin{cases} B^{n-3n/(2n+10)} (\log B)^2 & n\geq 8; \\ B^{n-n(n-2)/(6n+4)} (\log B)^2 & 2\leq n \leq 8. \end{cases} NEWLINE\]NEWLINE Applying this result Serre's conjecture is proved for cyclic covers of any degree for \(n\geq 10\) and even surpasses it for \(n>10\) and \(r\geq 3\).