Studying Hilbert's 10th problem via explicit elliptic curves

2022/07/14 by Debanjana Kundu, Antonio Lei, Kundu, Debanjana +3 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2207.07021

openalex publication_date 2022/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

N.García-Fritz and H.Pasten showed that Hilbert's 10th problem is unsolvable in the ring of integers of number fields of the form ℚ(√[3]p,√(-q)) for positive proportions of primes p and q. We improve their proportions and extend their results to the case of number fields of the form ℚ(√[3]p,√(Dq)), where D belongs to an explicit family of positive square-free integers. We achieve this by using multiple elliptic curves, and replace their Iwasawa theory arguments by a more direct method.

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