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arXiv · 1311.0043

Szpiro's small points conjecture for cyclic covers

Ariyan Javanpeykar·Rafael von Känel
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Abstract

Let $X$ be a smooth, projective and geometrically connected curve of genus at least two, defined over a number field. In 1984, Szpiro conjectured that $X$ has a "small point". In this paper we prove that if $X$ is a cyclic cover of prime degree of the projective line, then $X$ 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.

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Ariyan Javanpeykar, Rafael von Känel. 2014-03-23. Szpiro's small points conjecture for cyclic covers. https://arxiv.org/abs/1311.0043

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