A Survey of Local and Global Pairings on Elliptic Curves and Abelian Varieties

There are many bilinear pairings that naturally appear when one studies elliptic curves, abelian varieties, and related groups. Some of these pairings, notably the Weil and Lichtenbaum-Tate pairings, can be defined over finite fields and have important applications in cryptography. Others, such as the Neron-Tate canonical height pairing and the Cassels-Tate pairing on the Shafarevich-Tate group, are of fundamental importance in number theory and arithmetic geometry, but have seen limited use in cryptography. In this article I will present a survey of some of the pairings that are used to study elliptic curves and abelian varieties. I will attempt to show why they are natural pairings and how they fit into a wider framework.

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