I‑AIMS Institute for Artificial Intelligence
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Rational Points and Zeta Functions of Humbert Surfaces over F_q

Jorge Mello, Sajad Salami, Elira Shaska, Tanush Shaska

I-AIMS paper sh-91 · Published in NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur. 66, 92–122 · 2025

This paper examines the arithmetic of the loci ℒₙ, parameterizing genus 2 curves with (n,n)-split Jacobians over finite fields 𝔽_q. We compute rational points |ℒₙ(𝔽_q)| over 𝔽₃, 𝔽₉, 𝔽₂₇, 𝔽₈₁, and 𝔽₅, 𝔽₂₅, 𝔽₁₂₅, derive zeta functions Z(ℒₙ, t) for n = 2, 3. Utilizing these findings, we explore isogeny-based cryptography, introducing an efficient detection method for split Jacobians via explicit equations, enhanced by endomorphism ring analysis and machine learning optimizations. This advances curve selection, security analysis, and protocol design in post-quantum genus 2 systems, addressing efficiency and vulnerabilities across characteristics.

MSC classes: Primary 11G10; Secondary 11G25, 11G20, 11T71, 94A60

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