Kummer Surfaces, Isogenies and Theta Functions

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.