Sesquilinear pairings on elliptic curves

TL;DR

This paper introduces a generalized class of sesquilinear pairings on elliptic curves with complex multiplication, extending classical pairings like Weil and Tate-Lichtenbaum to broader algebraic contexts.

Contribution

It defines sesquilinear pairings on elliptic curves with complex multiplication, generalizing existing pairings to new algebraic structures involving rings like orders in quadratic fields or quaternion algebras.

Findings

Generalized pairings encompass Weil and Tate-Lichtenbaum pairings

New algebraic properties of sesquilinear pairings are established

Potential applications in cryptography and number theory are suggested

Abstract

Let $E$ be an elliptic curve with complex multiplication by a ring $R$, where $R$ is an order in an imaginary quadratic field or quaternion algebra. We define sesquilinear pairings ($R$-linear in one variable and $R$-conjugate linear in the other), taking values in an $R$-module, generalizing the Weil and Tate-Lichtenbaum pairings.