Hilbert’s 10th problem via Mordell curves

Authors

Somnath Jha
Debanjana Kundu, The University of Texas Rio Grande ValleyFollow
Dipramit Majumdar

Document Type

Article

Publication Date

2-26-2025

Abstract

We show that for 5 / 6 -th of all primes p, Hilbert’s 10th problem is unsolvable for the ring of integers of Q ( ζ 3 , 3 √ p ) . We also show that there is an infinite set S of square-free integers such that Hilbert’s 10th problem is unsolvable over the ring of integers of Q ( ζ 3 , √ D , 3 √ p ) for every D ∈ S and for every prime p ≡ 2 , 5 ( mod 9 ) . We use the CM elliptic curves y 2 = x 3 − 432 D 2 associated with the cube-sum problem, with D varying in suitable congruence class, in our proof.

Comments

© The Author(s), 2025.

Recommended Citation

Jha, Somnath, Debanjana Kundu, and Dipramit Majumdar. "Hilbert’s 10th problem via Mordell curves." Canadian Mathematical Bulletin (2024): 1-11. https://doi.org/10.4153/S0008439525000219

Publication Title

Canadian Mathematical Bulletin

DOI

10.4153/S0008439525000219