Analytic homomorphisms into Drinfeld modules
type: Item
instance of: scholarly article
title: Analytic homomorphisms into Drinfeld modules
zbMATH Open document ID: 0881.11055
DOI: 10.2307/2951814
author: Jing Yu
published in: Annals of Mathematics. Second Series
publication date: 1997-07-16T00:00:00Z
review text: Starting more than 10 years ago, the author has developed a remarkable theory of transcendence on Drinfeld modules (see for instance [\textit{Jing Yu}, Transcendence in finite characteristic, The arithmetic of function fields, de Gruyter, Berlin, 253-264 (1992; Zbl 0792.11015)]). The paper under review represents one of the most important steps in this theory. The author proves an analog of Philippon's zero estimate [\textit{P. Philippon}, Bull. Soc. Math. Fr. 114, No. 3, 355-383 (1986; Zbl 0617.14001)] and deduces an analog of Wüstolz's analytic subgroup theorem [\textit{G. Wüstholz}, Ann. Math., II. Ser. 129, No. 3, 501-517 (1989; Zbl 0675.10025)]. Denote by \(A\) the ring \({\mathbb{F}}_q[T]\) of polynomials over a finite field \({\mathbb{F}}_q\), by \(k\) the rational function field \({\mathbb{F}}_q(t)\) and by \(k_\infty\) its completion \({\mathbb{F}}_q((t))\). Let \(\overline k_\infty\) be an algebraic closure of \(k_\infty\) and \(\overline k\) be the algebraic closure of \(k\) in \(\overline k_\infty\). Let \(G=({\mathcal G}_{\text{a}}^m,\phi)\) be an \(m\)-dimensional \(t\)-module and \(\Phi:\overline k_\infty^n\rightarrow {\mathcal G}_{\text{a}}^m(\overline k_\infty)\) an \(n\)-parameters analytic submodule of \(G\). For each \(t\)-submodule \(H\) of \(G\), the non negative integer \(r(\Phi,H)\) is defined as the codimension in \(\overline k_\infty^n\) of \(d\Phi^{-1}(\text{Im } d\Phi\cap \text{Lie } H)\). For \(a\in k\) let \(d_\infty(a)\) denote the degree in \(t\) of \(a\). Given elements \(\gamma_1,\ldots,\gamma_\ell\) in \(G(\overline k_\infty)\) and a positive integer \(S\), define \[ \Gamma(S)=\bigl\{a_1\gamma_1+\cdots+a_\ell\gamma_\ell;\;a_i\in A,\;d_\infty(a_i)<S\bigr\}. \] Here is the zero estimate: Let \(Q\) be a polynomial in \(m\) variables which is of total degree \(\leq D\) and vanishes along \(\Phi\) to order \(\geq mT+1\) at all points of \(\Gamma(S)\). Then there exists a proper \(t\)-submodule \(H\subset G\) such that \[ {T+r(\Phi,H)\ch…
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11J81
zbMATH DE Number: 1014526
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: Carlitz modules
zbMATH Keywords: transcendence
zbMATH Keywords: linear independence
zbMATH Keywords: zero estimate
zbMATH Keywords: multiplicity estimate
zbMATH Keywords: analytic subgroup theorem
reviewed by: Michel Waldschmidt
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2951814
OpenAlex ID: W2321171465
description: scientific article; zbMATH DE number 1014526