Theta heights and Kodaira construction
type: Item
instance of: scholarly article
title: Theta heights and Kodaira construction
zbMATH Open document ID: 0947.14016
author: Gaël Rémond
published in: Journal of Number Theory
publication date: 2000-11-05T00:00:00Z
review text: The author proves that in the Kodaira construction a height on the base curve is described by the theta height of the jacobians of the fibres. He make use \ of theta heights as introduced by Bost and David who proved a comparison with Faltings' \ height. Therefore the author's result shows that a suitable effective version of the Shafarevich conjecture (a bound for Faltings' height) implies an effective Mordell. \ He gives a Kodaira construction with effectivity precisions and also provides a comparison between the above height and any given height on the curve. Theorem 3.1. Soient \(K\) un corps de nombres et \(S\) un ensemble fini de places finies de \(K\) contenant les places divisibles par \(2.\) Soit \(g\geq 2\) un entier et posons \(\gamma =4g-2.\) Notons \(B\) l'ouvert de \(\text{Spec }O_{K}\) obtenu en retirant les points correspondant aux places de \(S\) et \(\delta \) la valuation \(2\)-adique de l'exposant du groupe fini \(\text{Pic}(B).\) Soit \(L\) l'extension finie de \(K\) engendrée par toutes les extensions de \(K\) de degré au plus \( 2^{10g-3+\delta }\) non ramifiées en dehors de \(S.\) Alors pour toute courbe \(C\rightarrow K\) propre, lisse, connexe, de genre \(g, \) ayant bonne reduction en dehors de \(S\) et ayant un point rationnel sur \(K,\) il existe une courbe \(C''\) sur \(L\) propre, lisse, connexe et ayant un point rationnel sur \(L;\) un revêtement étale \(\alpha :C''\rightarrow C\times _{K}L\) de degré au plus \(2^{\gamma +1};\) un morphisme \(\pi :X\rightarrow C''\) propre et lisse, dont les fibres sont des courbes géométriquement intègres de genre \(\gamma ,\) qui est une fibration non isotriviale, admet une section \(\sigma :C''\rightarrow X\) et tel que pour tout point fermé \(Q\) de \(C''\) la fibre \(X_{Q}\) a bonne reduction en dehors de l'image réciproque de \(S\) dans les places du corps residuel \(k(Q).\)
Mathematics Subject Classification ID: 14H42
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14G40
zbMATH DE Number: 1375092
zbMATH Keywords: theta height
zbMATH Keywords: curves
zbMATH Keywords: abelian varieties
zbMATH Keywords: Jacobians
zbMATH Keywords: Shafarevich conjecture
zbMATH Keywords: bounds for height
zbMATH Keywords: effective Mordell
reviewed by: V. V. Chueshev
MaRDI profile type: Publication
full work available at URL: jnth.1999.2411
OpenAlex ID: W2074154238
cites work: Q3209230
cites work: Heights of Projective Varieties and Positive Green Forms
cites work: Q3727887
cites work: Q4006891
cites work: Finiteness theorems for abelian varieties over number fields.
cites work: Q5664012
cites work: Q5190299
cites work: Q3741712
cites work: Néron Models
cites work: Q5571531
cites work: Sur des hauteurs alternatives. I. (On alternative heights. I)
cites work: Q3715251
cites work: Q3999127
cites work: Siegel's theorem in the compact case
cites work: Q5609066
DOI: 10.1006/JNTH.1999.2411
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description: scientific article; zbMATH DE number 1375092