Siegel's lemma for function fields
type: Item
instance of: scholarly article
title: Siegel's lemma for function fields
zbMATH Open document ID: 0830.11024
author: Jeffrey Lin Thunder
published in: Michigan Mathematical Journal
publication date: 1996-02-01T00:00:00Z
review text: This interesting paper presents a function field analogue for the version of Thue-Siegel's lemma due to \textit{E. Bombieri} and \textit{J. Vaaler} [Invent. Math. 73, 11-32 (1983; Zbl 0533.10030)]. It is worthwhile to compare both the results and the methods. First assume that \(K\) is a number field of degree \(d\) over \(\mathbb{Q}\) with \(r_2\) distinct complex embeddings and discriminant \(\Delta_K\), and let \(M\) be an \(m\times n\) matrix of rank \(m\) with coefficients in \(K\). Bombieri and Vaaler show the existence of a basis \(x_1, \dots, x_{n- m}\) of the kernel of \(M\) in \(K^n\) such that \[ H(x_1) \dots H(x_{n-m})\leq \Biggl( {{2^{r_2} \sqrt {|\Delta_K |}} \over {\pi^{r_2}}} \Biggr)^{(n- m)/d} H' (M), \tag{1} \] where \(H(x)\) denotes the absolute multiplicative Weil height of a point \(x\in K^n\) computed using the maximum norm at each place, and where \(H' (M)\) denotes the absolute multplicative Weil height of the \({n\choose m}\)- tuple of minors of order \(m\) of \(M\) computed using the Euclidean norm at each archimedean place. Now assume that \(K\) is a function field in one variable over its constant field \(K_0\). More precisely, assume that \(K\) is an algebraic extension of \(K_0 (T)\) of degree \(d\) where \(T\in K\) is transcendental over \(K_0\). Let \(g\) be the genus of \(K\) and let \(M(K)\) be the set of all its valuations that are trivial on \(K_0\) and have value group \(\mathbb{Z}\). For each \(v\in M(K)\), let \(K_v\) denote the completion of \(K\) at \(v\) and, for each non-zero \(x\in (K_v )^n\), let \(\text{ord}_v (x)\) denote the minimum of the order at \(v\) of the \(n\) coordinates of \(x\). The author defines the divisor of a non-zero \(x\in K^n\) as \(\text{div} (x)= \sum_v \text{ord}_v (x)\) and the (logarithmic) height of \(x\) as \(h(x)= -(1/ d)\deg (\text{div} (x))\). For a matrix \(M\) as above, he shows the existence of a basis \(x_1, \dots, x_{n-m}\) of the kernel of \(M\) in \(K^n\) such that \[ h(x_1)+ \cdots+ …
reviewed by: Damien Roy
Mathematics Subject Classification ID: 11J13
Mathematics Subject Classification ID: 11D04
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R56
zbMATH DE Number: 784589
zbMATH Keywords: logarithmic height
zbMATH Keywords: function field analogue
zbMATH Keywords: Thue-Siegel's lemma
zbMATH Keywords: Weil height
zbMATH Keywords: adeles
zbMATH Keywords: Riemann-Roch theorem
MaRDI profile type: Publication
Wikidata QID: Q124816303
full work available at URL: https://doi.org/10.1307/mmj/1029005160
OpenAlex ID: W2015770859
DOI: 10.1307/MMJ/1029005160
description: scientific article; zbMATH DE number 784589