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Computation of genus 2 Kleinian hyperelliptic functions via Richelot isogenies

Matvey Smirnov

Abstract

In this work we propose an algorithm that numerically evaluates Kleinian hyperelliptic functions associated with a complex curve of genus 2. This algorithm is based upon constructing a sequence of curves with Richelot isogenous Jacobians and a recurrent procedures that reduces the calculation to a degenerate curve. As a part of mentioned algorithm we propose a method of choosing a Richelot isogenous curve (among 15 possibilities) that guarantees convergence of the equations of the curves and associated Kleinian functions of weight 2 under iterations.

Computation of genus 2 Kleinian hyperelliptic functions via Richelot isogenies

Abstract

In this work we propose an algorithm that numerically evaluates Kleinian hyperelliptic functions associated with a complex curve of genus 2. This algorithm is based upon constructing a sequence of curves with Richelot isogenous Jacobians and a recurrent procedures that reduces the calculation to a degenerate curve. As a part of mentioned algorithm we propose a method of choosing a Richelot isogenous curve (among 15 possibilities) that guarantees convergence of the equations of the curves and associated Kleinian functions of weight 2 under iterations.
Paper Structure (13 sections, 18 theorems, 96 equations)

This paper contains 13 sections, 18 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Curves of genus 2 and Kleinian functions
  4. Operations with polynomials of degree 2
  5. Richelot isogeny
  6. Iterations of the Richelot construction
  7. Polynomials subordinate to a triple of disks
  8. Convergence of the iterations of the Richelot construction
  9. Limiting values of Kleinian functions
  10. Algorithms and numerical experiments
  11. Algorithm to compute periods.
  12. Algorithm to compute values of Kleinian functions
  13. Algorithm to compute the Abel map

Key Result

Proposition 2.1

The following statements hold for all non-zero $p,q \in \mathfrak P_2$.

Theorems & Definitions (35)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Remark
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • ...and 25 more