Weil pairing for Drinfeld modules
type: Item
instance of: scholarly article
title: Weil pairing for Drinfeld modules
zbMATH Open document ID: 1077.11044
author: Gert-Jan van der Heiden
published in: Monatshefte für Mathematik
publication date: 2005-02-07T00:00:00Z
review text: The main goal of this paper is the construction for Drinfeld modules of an analogue of Weil pairing for elliptic curves. Let \(K\) be a global function field and \(A\) the ring of functions regular outside of a fixed place \(\infty\). In sections 1--3 the author proves some basic results about (Drinfeld-Anderson) \(A\)-motives (cf. also [\textit{I. Potemine}, ''Drinfeld-Anderson motives and multicomponent KP hierarchy''. Proceedings of an international conference, KAIST, Taejon, South Korea, August 11--15, 1997. Providence, RI: American Mathematical Society. Contemp. Math. 224, 213--227 (1999; Zbl 0926.11041)]). For any ideal \(I\subset A\), denote \(\Omega_I=I^{-1}\Omega_A/\Omega_A\) where \(\Omega_A\) is the module of differentials of \(A\). It is easy to show that the \(A\)-module \(P_I\) of \(I\)-torsion points of any \(A\)-motive of rank \(r\) is isomorphic to \((A/I)^r\) (corollary 4.4). The notion of determinant of an \(A\)-motive permits to define a map \(w_I:\prod_{i=1}^r P_I\rightarrow P_I\otimes \Omega_I^{\otimes r-1}\) which is an analogue of Weil pairing (theorem 5.3). In the final section, the author gives an explicit example in the case \(A=\mathbb{F}_q[T]\), \(r=2\).
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 2132187
zbMATH Keywords: \(A\)-motive
zbMATH Keywords: Drinfeld module
zbMATH Keywords: global function field
zbMATH Keywords: Weil pairing
reviewed by: Igor Potemine
MaRDI profile type: Publication
full work available at URL: s00605-004-0261-4
OpenAlex ID: W2145926043
DOI: 10.1007/S00605-004-0261-4
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description: scientific article; zbMATH DE number 2132187