Effective calculation of the geometric height and the Bogomolov conjecture for hyperelliptic curves over function fields

Yamaki, Kazuhiko

doi:10.1215/kjm/1250271420

Cited by 19 publications

(27 citation statements)

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“…The definition may seem rather ad hoc at first sight, but in the function field context the invariant already occurs, as mentioned before, in work of A. Moriwaki [13] and K. Yamaki [16]. In the next section we present a more intrinsic approach to χ, using the arithmetic of symmetric roots.…”

Section: The Invariant χmentioning

confidence: 98%

“…In the case g ≥ 3 one has an effective lower bound for χ(X) which is strictly positive in the case of non-smooth reduction, by work of Yamaki [16]. We quote his result:…”

Section: ωη and D(x)mentioning

confidence: 99%

“…Assume that X has semistable reduction over K. We study, for each place v of K, a real-valued invariant χ(X v ) of X ⊗ K v , with the following two properties: In the function field context, the invariant χ already appears in work of A. Moriwaki [13] and K. Yamaki [16], albeit in disguise. It follows from their work that χ(X v ) is strictly positive if X has non-smooth reduction at v. In fact they prove a precise lower bound for χ(X v ) in terms of the geometry of the special fiber at v. If X has a non-isotrivial model, by property (ii) this yields as a corollary an effective proof of the Bogomolov conjecture for X, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Symmetric roots and admissible pairing

Jong

2011

Trans. Amer. Math. Soc.

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Abstract. Using the discriminant modular form and the Noether formula it is possible to write the admissible self-intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum, over all places, of a certain adelic invariant χ. We provide a simple geometric interpretation for this invariant χ, based on the arithmetic of symmetric roots. We propose the conjecture that the invariant χ coincides with the invariant ϕ introduced in a recent paper by S.-W. Zhang. This conjecture is true in the genus 2 case, and we obtain a new proof of the Bogomolov conjecture for curves of genus 2 over number fields.

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Section: The Invariant χmentioning

confidence: 98%

“…In the case g ≥ 3 one has an effective lower bound for χ(X) which is strictly positive in the case of non-smooth reduction, by work of Yamaki [16]. We quote his result:…”

Section: ωη and D(x)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Symmetric roots and admissible pairing

Jong

2011

Trans. Amer. Math. Soc.

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“…Whenever f is smooth, we clearly have ω 2 a = ω 2 X/Y by Equation (2), and that ω 2 X/Y ≥ 12 as Paršin [P] showed. The Effective Bogomolov Conjecture (Conjecture 2.2) was known to be true for curves of genus less than 5 ( [AM1], [AM2], [AM3], [AM4], [KY1], [KY2], and [Fa]). Also, W. Gubler [G] showed that the Bogomolov Conjecture is true for C if the Jacobian variety of C has totally degenerate reduction over some point y ∈ Y .…”

Section: Conjecture 22 [Ky1](effective Bogomolovmentioning

confidence: 99%

“…Following Zhang's approach, A. Moriwaki used metrized graphs and Green's functions to prove specific cases of Bogomolov's conjecture over function fields in a series of papers, [AM1], [AM2], and [AM3]. Extending Moriwaki's approach, Yamaki [KY1] proved very special cases of effective generalized Bogomolov's conjecture over function fields.…”

Section: Metrized Graphs and Their Tau Constantsmentioning

confidence: 99%

Zhang’s Conjecture and the Effective Bogomolov Conjecture over function fields

Cinkir

2010

Invent. math.

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Abstract. We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang's Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of genus at most 4 and a few other special cases. We also either verify or improve the previous results. We relate the invariants involved in Zhang's Conjecture to the tau constant of metrized graphs. Then we use and extend our previous results on the tau constant. By proving another Conjecture of Zhang, we obtain a new proof of the slope inequality for Faltings heights on moduli space of curves.

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Gross–Schoen cycles and dualising sheaves

Zhang

2009

Invent. math.

191

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Effective calculation of the geometric height and the Bogomolov conjecture for hyperelliptic curves over function fields

Cited by 19 publications

References 0 publications

Symmetric roots and admissible pairing

Symmetric roots and admissible pairing

Zhang’s Conjecture and the Effective Bogomolov Conjecture over function fields

Gross–Schoen cycles and dualising sheaves

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