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Sesquilinear pairings on elliptic curves

Katherine E. Stange

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.

Sesquilinear pairings on elliptic curves

Abstract

Let be an elliptic curve with complex multiplication by a ring , where is an order in an imaginary quadratic field or quaternion algebra. We define sesquilinear pairings (-linear in one variable and -conjugate linear in the other), taking values in an -module, generalizing the Weil and Tate-Lichtenbaum pairings.
Paper Structure (17 sections, 17 theorems, 148 equations, 1 algorithm)

This paper contains 17 sections, 17 theorems, 148 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Classical pairings
  3. The Weil pairing
  4. The Tate-Lichtenbaum pairing
  5. The calculus of $R$-divisors
  6. $R$-divisors
  7. Evaluation of functions at divisors
  8. Weil reciprocity
  9. Sesquilinear pairings
  10. Generalization of Tate-Lichtenbaum pairing
  11. Generalization of Weil pairing
  12. Curves with complex multiplication
  13. Pull-back to CM curves
  14. Quadratic case
  15. Pairings for quadratic $R$
  16. ...and 2 more sections

Key Result

Proposition 2.3

Suppose $m$ is coprime to $\operatorname{char}(K)$ in the case of positive characteristic. Definitions defn: weil1 and defn: weil2 are well-defined, equal when defined, and have the following properties (where defined in the case of the first definition):

Theorems & Definitions (43)

  • Definition 2.1: Weil pairing: first definition
  • Definition 2.2: Weil pairing: second definition
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Theorem 3.1
  • proof
  • ...and 33 more