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Elliptic normal curves of even degree and theta functions

Masanobu Kaneko, Masato Kuwata

Abstract

An elliptic curve may be immersed in ${\mathbf{P}}^{N-1}$ as a degree $N$ curve using level $N$ structure. In the case where $N$ is odd, there are well known classical results dating back to Bianchi and Klein. In this paper we study the case of even $N$ in some detail. In particular, over the complex number field, we define an immersion using suitably chosen theta functions, and study the quadratic equations satisfied by them.

Elliptic normal curves of even degree and theta functions

Abstract

An elliptic curve may be immersed in as a degree curve using level structure. In the case where is odd, there are well known classical results dating back to Bianchi and Klein. In this paper we study the case of even in some detail. In particular, over the complex number field, we define an immersion using suitably chosen theta functions, and study the quadratic equations satisfied by them.

Paper Structure

This paper contains 22 sections, 37 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Weil pairing
  4. Central extension $E[N](D)$
  5. Projective immersion associated with a cyclic subgroup
  6. Canonical coordinate system
  7. Odd case
  8. Even case
  9. Theta functions
  10. Basic properties of $\theta^{(N)}_{k}(z,\tau)$ as a function of $z$
  11. Projective immersion via theta functions
  12. Transformation formula for $\theta^{(N)}_{k}(z,\tau)$
  13. Modular curves
  14. Quadratic equations satisfied by theta functions
  15. Odd case
  16. ...and 7 more sections

Key Result

Theorem \oldthetheorem

The following sequence is exact.

Theorems & Definitions (87)

  • Theorem \oldthetheorem: Abel-Jacobi
  • Theorem \oldthetheorem: Weil pairing
  • proof
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem: Vélu Velu:1978
  • Lemma \oldthetheorem: Vélu Velu:1978
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
  • ...and 77 more