Kummer Surfaces, Isogenies and Theta Functions

2025/05/19 by Clingher, Adrian, Malmendier, Andreas, Shaska, Tony · 1 citation
#14J28 #14K02 #14K25 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.13727

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.

Citations

• Rational Points and Zeta Functions of Humbert Surfaces with Square Discriminant
• Hash functions from superspecial genus-2 curves using Richelot isogenies
• Lattice Polarized K3 Surfaces and Siegel Modular Forms
• Thetanulls of cyclic curves of small genus
• Curves of genus 2 with (n, n)-decomposable jacobians

Cited by

• Graded Transformers

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• Isogenies on Kummer Surfaces
• Efficient (3,3)-isogenies on fast Kummer surfaces
• Computing isogeny classes of typical principally polarized abelian surfaces over the rationals
• Spanning isogeny classes of principally polarized abelian surfaces with RM
• Theta characteristics and the fixed locus of [-1] on some varieties of Kummer type