JournalsdmVol. 19pp. 1085–1103

Szpiro's small points conjecture for cyclic covers

  • Ariyan Javanpeykar

    Mathematical Institute IHÉS University of Leiden 35 Route de Chartres 2300RA Leiden 91440 Bures-sur-Yvette The Netherlands France Current address: Current address: Institut für Mathematik MPIM Bonn J.G.-Universität Mainz Vivatsgasse 7 55099 Mainz 53111 Bonn Germany Germany

  • Rafael von Känel

    Mathematical Institute IHÉS University of Leiden 35 Route de Chartres 2300RA Leiden 91440 Bures-sur-Yvette The Netherlands France Current address: Current address: Institut für Mathematik MPIM Bonn J.G.-Universität Mainz Vivatsgasse 7 55099 Mainz 53111 Bonn Germany Germany
Szpiro's small points conjecture for cyclic covers cover
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Abstract

Let be a smooth, projective and geometrically connected curve of genus at least two, defined over a number field. In 1984, Szpiro conjectured that has a «small point». In this paper we prove that if is a cyclic cover of prime degree of the projective line, then has infinitely many «small points». In particular, we establish the first cases of Szpiro's small points conjecture, including the genus two case and the hyperelliptic case. The proofs use Arakelov theory for arithmetic surfaces and the theory of logarithmic forms.

Cite this article

Ariyan Javanpeykar, Rafael von Känel, Szpiro's small points conjecture for cyclic covers. Doc. Math. 19 (2014), pp. 1085–1103

DOI 10.4171/DM/475