Weil pairing for Drinfeld modules
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 \(A=\mathbb{F}_q[T]\), \(r=2\).
scientific article; zbMATH DE number 2132187