Kummer Surfaces, Isogenies and Theta Functions

TL;DR

This paper explores the geometric and computational aspects of $(n,n)$-isogenies for Abelian surfaces and Kummer surfaces, utilizing Theta functions to classify genus-two curves and develop efficient algorithms for isogeny computations.

Contribution

It provides a comprehensive Theta function framework for classifying genus-two curves and extends Richelot's isogeny methods to higher orders for improved computational efficiency.

Findings

Explicit quartic normal forms for Kummer surfaces derived from Theta functions

Practical algorithms for $(n,n)$-isogeny computations developed

Extensions of Richelot's $(2,2)$-isogenies to higher order cases proposed

Abstract

The paper discusses geometric and computational aspects associated with $(n,n)$-isogenies for principally polarized Abelian surfaces and related Kummer surfaces. We start by reviewing the comprehensive Theta function framework for classifying genus-two curves, their principally polarized Jacobians, as well as for establishing explicit quartic normal forms for associated Kummer surfaces. This framework is then used for practical isogeny computations. A particular focus of the discussion is the $(n,n)$-Split isogeny case. We also explore possible extensions of Richelot's $(2,2)$-isogenies to higher order cases, with a view towards developing efficient isogeny computation algorithms.