Hilbert's 10th Problem via Mordell curves

2024/12/05 by Somnath Jha, Jha, Somnath, Debanjana Kundu +3
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2412.04253

openalex publication_date 2024/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that for 5/6-th of all primes p, Hilbert's 10-th Problem is unsolvable for ℚ(ζ₃, √[3]p). We also show that there is an infinite set S of square free integers such tha Hilbert's 10-th Problem is unsolvable over the number fields ℚ(ζ₃, √(D), √[3]p) for every D ∈ S and every prime p ≡ 2,5 \pmod9. We use the CM elliptic curves Y²=X³-432D² associated to the cube sum problem, with D varying in suitable congruence class, in our proof.

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