Szpiro's small points conjecture for cyclic covers
The paper under review proves Szpiro's small points conjecture for cyclic covers of \(\mathbb P^1\) of prime order.NEWLINENEWLINELet \(K\) be a number field. In the paper, a \textit{curve} over \(K\) is defined to be a geometrically connected, smooth and projective curve \(X\to\text{Spec} K\). The height \(h(x)\) of a point \(x\in X(\overline k)\) is the Arakelov height, obtained by intersecting with the dualizing sheaf on a semistable arithmetic surface associated to \(X\) (over the ring of integers of a suitable finite extension of \(K\)). The Néron-Tate height \(h_{\text{NT}}(x)\) is the Néron-Tate height of the point \((2g-2)x-K\) on the Jacobian of \(X\), where \(K\) is a canonical divisor of \(X\). Both heights are absolute logarithmic heights.NEWLINENEWLINEThe statement of the main theorem uses the following notation. Let \(g\geq2\) be an integer, let \(K\) be a number field, and let \(S\) be a finite set of non-archimedean places of \(K\). Let \(D_K\) be the absolute value of the discriminant of \(K\) over \(\mathbb Q\), let \(d=[K:\mathbb Q]\), and let \(N_S=\prod_{v\in S} N_v\), where \(N_v\) is the number of elements in the residue field at \(v\). For brevity, let \(\nu=d(5g)^5\).NEWLINENEWLINEThen the main theorem says the following. Let \(X\) be a curve over \(K\) of genus \(g\). Let \(S\) be the set of non-archimedean places of bad reduction for \(X\). Assume that there is a finite morphism \(X\to\mathbb P^1_K\) of prime degree which is geometrically a cyclic cover. Then there exist infinitely many points \(x\in X(\overline K)\) that satisfy NEWLINE\[NEWLINE\log\max(h_{\text{NT}}(x),h(x)) \leq \nu^{d\nu}(N_SD_K)^\nu\;.NEWLINE\]NEWLINENEWLINENEWLINEVariants of the theorem, with sharper bounds, are also proved in the special cases of hyperelliptic curves and curves of genus \(2\).NEWLINENEWLINEThe proofs use Arakelov theory of arithmetic surfaces, degree bounds for Belyi maps due to \textit{L. S. Khadjavi} [J. Number Theory 96, No. 1, 22--47 (2002; Zbl 1078.11046)], combined with work of \textit{R. de Jong} and \textit{G. Rémond} [Algebra Number Theory 5, No. 8, 1133--1143 (2011; Zbl 1276.11110)] and \textit{R. von Känel} [Int. Math. Res. Not. 2014, No. 16, 4457--4491 (2014; Zbl 1321.11070)], which relies on Baker's theory of logarithmic forms.
scientific article; zbMATH DE number 6359331