What does this research mean for the field?

A positive density of elliptic curves defined over Q is diophantine stable in certain Galois extensions. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Investigate the density of elliptic curves that are diophantine stable over Galois extensions of rational numbers.

February 21, 2026

A positive density of elliptic curves is diophantine stable in certain Galois extensions

Key Points

Investigate the density of elliptic curves that are diophantine stable over Galois extensions of rational numbers.
Consider cyclic p-extension L/Q for p in {3,5}
Establish criteria for diophantine stability
Analyze elliptic curves ordered by height
Demonstrated an effective positive density of diophantine stable elliptic curves.
Identified specific elliptic curves defined over Q that meet the stability criteria.

Abstract

Abstract Let p ∈ 3, 5 p\{3, 5\} and consider a cyclic p -extension L / ℚ L/Q. We show that there exists an effective positive density of elliptic curves E defined over ℚ Q, ordered by height, that is diophantine stable in L.

A positive density of elliptic curves is diophantine stable in certain Galois extensions

Key Points

Abstract

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