Rational points on families of curves with genus at least 2
type: Item
instance of: scholarly article
title: Rational points on families of curves with genus at least 2
zbMATH Open document ID: 0896.11025
author: Teresa de Diego
published in: Journal of Number Theory
publication date: 1997-12-18T00:00:00Z
review text: Let \({\mathcal C}\) and \({\mathcal T}\) be two projective varieties defined over a number field \(k\). Suppose that there exists a flat morphism \(\pi:{\mathcal C} \to {\mathcal T}\) of \(k\)-schemas such that the generic fibre \({\mathcal C}_\eta ={\mathcal C} \times_{\mathcal T} \text{Spec} ({\mathcal T})\) is a smooth curve of genus \(\geq 2\) over \(k(\eta) \cong k({\mathcal T})\). We denote by \({\mathcal T}^0\) the open of \({\mathcal T}\) over which the morphism \(\pi\) is smooth. For every \(t\in {\mathcal T}^0 (\overline k)\) we consider the curve \({\mathcal C}_t\) embedded in its Jacobian \({\mathcal J}_t\) and we denote by \(| \cdot|\) the Euclidean norm associated to the theta divisor. In this paper the author proves that there exist two positive constants \(\gamma_1 ({\mathcal C})\) and \(\gamma_2 (g)\) such that for every point \(t\in {\mathcal T}^0 (\overline k)\) and every finite extension \(k'\) of \(k(t) \) we have \[ \text{Card} \biggl\{x\in {\mathcal C}_t(k')/ | x| \geq\gamma_1 ({\mathcal C}) \bigl( \sqrt {h_\tau (t)}+1 \bigr)\biggr\} \leq\gamma_2 (g)7^{\text{rank} {\mathcal J}_t (k')}\leq {55\over 2} 7^{\text{rank} {\mathcal J}_t (k')}. \] The constant \(\gamma_1 ({\mathcal C})\) depends only on \({\mathcal C}\) and the choice of the height \(h_\tau\) over \({\mathcal T}\) associated to an ample divisor \(\tau\) and the constant \(\gamma_2 (g)\) is a decreasing function of \(g\). This result is a uniform version of the well-known Falting's theorem which asserts that every algebraic curve of genus \(\geq 2\) defined over a number field has a finite number of rational points. The proof of this theorem by Vojta and Bombieri uses the inequalities of Mumford and Vojta [\textit{E. Bombieri}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, 615--640 (1990; Zbl 0722.14010); ibid. 18, 473 (1991; Zbl 0763.14007)]. The author, using Vojta-Bombieri methods, proves uniform versions of the inequalities of Mumford and Vojta which imply its result. As an a…
reviewed by: Dimitros Poulakis
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 1097549
zbMATH Keywords: theta divisor
zbMATH Keywords: Neron-Tate height
zbMATH Keywords: arbitrary genus
zbMATH Keywords: projective varieties
zbMATH Keywords: Falting's theorem
zbMATH Keywords: algebraic curve
zbMATH Keywords: rational points
zbMATH Keywords: Szpiro's conjecture
zbMATH Keywords: jacobians
zbMATH Keywords: elliptic curves
MaRDI profile type: Publication
full work available at URL: jnth.1997.2146
OpenAlex ID: W1992638917
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DOI: 10.1006/JNTH.1997.2146
description: scientific article; zbMATH DE number 1097549