Skew-symmetric pairing on polynomial formal modules

Abstract

Let K be a local field, O_K be the ring of integers of K, c be a unit in O_K, and F_c (X, Y) = X + Y + cXY be a polynomial formal group law. Let F_c(m_K) be the formal module defined by F_c(X, Y) on the maximal ideal of O_K and [pⁿ]F_c (X) be an isogeny of F_c. In the present paper, we give an explicit construction of a pairing (,)_c: F_c(m_K) × F_c(m_K) → Ker[pⁿ]F_c generalizing the classical Hilbert pairing.