Class pairings and isogenies on elliptic curves

Abstract

In [Bue76,Bue77] a non-trivial homomorphism δ was constructed from Q-rational points on an elliptic curve to the ideal class group of a quadratic field K=Q(D). In [MT83] it was conjectured that δ was related to a pairing on points of E(K), which has come to be known as the ideal class pairing. In this paper we will show how to write the ideal class pairing in an explicit and easy-to-compute manner, and we will prove by direct calculation that the homomorphism δ is identical to the ideal class pairing with one argument fixed. For any elliptic curve E over a number field K, we will show how to extend the ideal class pairing to a full pairing on E(K), called simply the elliptic curve class pairing, which maps into an extension of the idele class group of K. The explicit formulas we will derive for the elliptic curve class pairing will enable us to prove that there is a natural relationship between this pairing and the Weil descent pairing. The kernel of the elliptic curve class pairing will be determined in many cases where one argument is taken to be a fixed torsion point and, as a corollary, the kernel of δ will be computed for a class of elliptic curves over imaginary quadratic fields.