Research reveals a positive density of elliptic curves that exhibit diophantine stability in specific Galois extensions, suggesting important implications for number theory.
Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mn>5</m:mn> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {p∈\{3,5\}} and consider a cyclic p -extension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>L</m:mi> <m:mo>/</m:mo> <m:mi>ℚ</m:mi> </m:mrow> </m:math> {L/Q} . We show that there exists an effective positive density of elliptic curves E defined over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℚ</m:mi> </m:math> {Q} , ordered by height, that is diophantine stable in L .
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Ray et al. (2026) studied this question.
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