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An explicit expression of the Richelot isogeny through Kleinian hyperellyptic functions

Matvey Smirnov

Abstract

We find an explicit expression for the Richelot isogeny of Kummer surfaces of genus 2 curves in terms of Kleinian hyperelliptic functions of weight 2. We use this expression to relate Kleinian hyperelliptic functions associated to Richelot isogenous curves.

An explicit expression of the Richelot isogeny through Kleinian hyperellyptic functions

Abstract

We find an explicit expression for the Richelot isogeny of Kummer surfaces of genus 2 curves in terms of Kleinian hyperelliptic functions of weight 2. We use this expression to relate Kleinian hyperelliptic functions associated to Richelot isogenous curves.
Paper Structure (11 sections, 12 theorems, 83 equations)

This paper contains 11 sections, 12 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Curves of genus 2 and Kleinian hyperelliptic functions
  4. Richelot construction
  5. Kummer surfaces
  6. Richelot isogeny of Kummer surfaces
  7. General form of the Richelot isogeny of Kummer surfaces
  8. Special properties of the components of mathfrak R (p,q,r)
  9. Explicit expression for the Richelot isogeny
  10. Calculation of the rows of matrices mathfrak B and hat mathfrak B
  11. Relation between Kleinian functions associated with Richelot isogenous curves

Key Result

Proposition 2.1

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

Theorems & Definitions (27)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Lemma 3.1
  • proof
  • Remark
  • ...and 17 more