Arithmetic of elliptic curves induced by regular Diophantine triples

Abstract

We study elliptic curves induced by regular Diophantine triples, with emphasis on their torsion subgroups. We show that an elliptic curve $E$ induced by a regular Diophantine triple in integers necessarily has torsion subgroup $E(\mathbb{Q})_{\mathrm{tors}} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. Moreover, we develop a criterion for when such an elliptic curve acquires a point of order $3$ over a quadratic field. For a particular family $\{ k-1, k+1, 4k\}$, we use it to show that this does not happen. Finally, we study both the torsion and the generic rank of a family of elliptic curves induced by the $D(-k^2)$-triple $\{1, 2k^2, 2k^2+2k+1\}$.

Disclosure

“such that the induced elliptic curve has Z/2Z×Z/6Z-torsion subgroup over Q? Let us note that this last question also appears in Dujella’s extensive list of open problems on Diophantine m-tuples and elliptic curves [Duj26]. Declaration of generative AI in the manuscript preparation The proofs of Theorem 12 and Lemma 11 were AI-assisted: this required work with ChatGPT and Gemini in many iterations; at the time (January-February 2026), neither was capable of generating the proofs independ”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T19:18:18.005757Z

Structural counts