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arXiv · 2412.04253

Hilbert's 10th Problem via Mordell curves

Somnath Jha·Debanjana Kundu·Dipramit Majumdar
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Abstract

We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(\zeta_3, \sqrt[3]{p})$. 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 $\mathbb{Q}(\zeta_3, \sqrt{D}, \sqrt[3]{p})$ for every $D \in S$ and every prime $p \equiv 2,5 \pmod{9}$. We use the CM elliptic curves $Y^2=X^3-432D^2$ associated to the cube sum problem, with $D$ varying in suitable congruence class, in our proof.

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Somnath Jha, Debanjana Kundu, Dipramit Majumdar. 2024-12-05. Hilbert's 10th Problem via Mordell curves. https://arxiv.org/abs/2412.04253

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