I‑AIMS Institute for Artificial Intelligence
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Isogenies, Kummer surfaces, and theta functions

Adrian Clingher, Andreas Malmendier, Tanush Shaska

I-AIMS paper sh-93 · Published in NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur. 66, 1–55 · 2025

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.

MSC classes: 14J28, 14K02, 14K25

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