Experimental Mathematics

The Geometric Bogomolov Conjecture for Curves of Small Genus

X. W. C. Faber

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Abstract

The Bogomolov conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov conjecture for all curves of genus at most $4$ over a function field of characteristic zero. We recover the known result for genus-$2$ curves and in many cases improve upon the known bound for genus-$3$ curves. For many curves of genus $4$ with bad reduction, the conjecture was previously unproved.

Article information

Source
Experiment. Math., Volume 18, Issue 3 (2009), 347-367.

Dates
First available in Project Euclid: 25 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.em/1259158471

Mathematical Reviews number (MathSciNet)
MR2555704

Zentralblatt MATH identifier
1186.11035

Subjects
Primary: 11G30: Curves of arbitrary genus or genus = 1 over global fields [See also 14H25]
Secondary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30] 11G50: Heights [See also 14G40, 37P30]

Keywords
Bogomolov conjecture curves of higher genus function fields metric graphs

Citation

Faber, X. W. C. The Geometric Bogomolov Conjecture for Curves of Small Genus. Experiment. Math. 18 (2009), no. 3, 347--367. https://projecteuclid.org/euclid.em/1259158471


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