A certain property of the Hilbert pairing
Let \(k_ 0\) be a local field of characteristic zero, i.e. a finite algebraic extension of the field \(\mathbb Q_ p\) of \(p\)-adic integers, \({\mathfrak O}\) its ring of integers and \(\pi\) its uniformizing element. Let \(F\) be a Lubin-Tate one-dimensional formal group over \({\mathfrak O}\) attached to \(\pi\) [\textit{J. Lubin} and \textit{J. Tate}, Ann. Math. (2) 81, 380--387 (1965; Zbl 0128.26501)]. Let \(n\) be a positive integer, \(\kappa_ n\) the kernel of the endomorphism \([\pi]^ n\) of \(F\) and \(k_ n=k_ 0(\kappa_ n)\) the field of definition of \(\kappa_ n\) (ibid.). We write \({\mathfrak m}\) for the maximal ideal of the ring of integers of \(k_ n\). One may view \({\mathfrak m}\) as a commutative group \(F({\mathfrak m})\) with respect to the group law of \(F\). The following result of the paper under review proves the non-existence of non-trivial generalized Hilbert symbols for all Lubin-Tate groups [cf. \textit{S. Lang}, ``Cyclotomic fields.'' New York etc.: Springer-Verlag (1978; Zbl 0395.12005); chapter 8]. Theorem. Let us assume that for all Lubin-Tate groups there is a pairing \(c_ F: k_ n\times F({\mathfrak m})\to \kappa_ n\) enjoying the following properties: (1) functoriality: if a formal series \(\phi(X)\) defines an isomorphism of Lubin-Tate groups \(F\) and \(G\) then \(\phi c_ F(a,b)=c_ G(a,\phi (b));\) (2) Hilbert symbol fundamental property: \(c_ F(a,-a)=0\) for all \(a\) from \({\mathfrak m}.\) Then \(c_ F=0\).
scientific article; zbMATH DE number 4132345
scientific article