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Hilbert’s 10th problem via Mordell curves
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 26 February 2025, pp. 1-11
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- Article
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We show that for
$5/6$-th of all primes p, Hilbert’s 10th problem is unsolvable for the ring of integers of 
$\mathbb {Q}(\zeta _3, \sqrt [3]{p})$. 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 
$\mathbb {Q}(\zeta _3, \sqrt {D}, \sqrt [3]{p})$ for every 
$D \in S$ and for every prime 
$p \equiv 2, 5\ \pmod 9$. 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.
