Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

QRT Map on a Bielliptic Surface

Nalini Joshi, Frank W. Nijhoff, Allan Steel

Abstract

The family of mappings of the plane possessing a biquadratic invariant, which is known collectively as QRT maps, is composed of two involutions, one preserving a vertical shift and the other preserving a horizontal shift in the plane. In this paper, we extend the map by replacing each shift by the group operation on each of two families of elliptic curves, whose product forms a bielliptic surface.

QRT Map on a Bielliptic Surface

Abstract

The family of mappings of the plane possessing a biquadratic invariant, which is known collectively as QRT maps, is composed of two involutions, one preserving a vertical shift and the other preserving a horizontal shift in the plane. In this paper, we extend the map by replacing each shift by the group operation on each of two families of elliptic curves, whose product forms a bielliptic surface.

Paper Structure

This paper contains 27 sections, 1 theorem, 48 equations.

Table of Contents

  1. Introduction
  2. bielliptic QRT
  3. Constant moduli case
  4. Geometry of the bielliptic QRT map
  5. Conclusion
  6. Special branches of solutions
  7. Case 1: $i_1 = 0$
  8. Case 1.1: $i_1 = 0$, $h_0h_1f_1\not=0$
  9. Case 1.2: $i_1 = 0$, $h_0=0$, with $e_0h_1f_1\not=0$
  10. Case 1.3: $i_1 = 0$,$h_0=0$, $f_1=0$, with $i_0\,h_1\,e_0\not=0$
  11. Case 1.4
  12. Case 2: $h_0 = 0$
  13. Case 2.1: $h_0=0$, $i_1\, e_0\,(e_1 i_1 - f_1 h_1)\not=0$
  14. Case 2.2: $h_0=0$, $(e_1 i_1 - f_1 h_1)=0$, $i_1\,h_1\,i_0 e_0\not=0$
  15. Case 2.3: $h_0=0$, $(e_1 i_1 - f_1 h_1)=0$, $h_1=0$, $i_1\, e_0\not=0$
  16. ...and 12 more sections

Key Result

Lemma 2.1

The map defined by eq:3det together with eq:Weier is an involution, i.e., $\widetilde{\widetilde{\boldsymbol{X}}}=\boldsymbol{X}$.

Theorems & Definitions (5)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Example 3.1