© Cambridge University Press 2015. The classical Hilbert symbol determines a skew-symmetric pairing on the multiplicative group of a local field. Explicit formulae for the Hilbert symbol were obtained independently by Brückner () and Vostokov () in 1978. Subsequently, these formulae were generalized to multi-dimensional local fields (see ). In this note, we shall generalize these results to the formal modules. Let F be a commutative formal group over the ring of integers of a local field k; let k0 ∩ k be the subfield such that is the endomorphism ring of F. Let k′|k be a finite extension and let, where π is a prime element of k′. Our goal is to construct a skew-symmetric bilinear pairing (α, β): F(M) × F(M) → ker[πn]F.
|Title of host publication||Arithmetic and Geometry|
|State||Published - 2015|