Let K be a number field and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\ell \geqslant 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>⩾</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> a prime number. Mazur and Rubin introduced the notion of diophantine stability for a variety <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$X_{/K}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mrow> <mo stretchy="false">/</mo> <mi>K</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> at a prime <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>. We show that there is a positive density set of elliptic curves <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$E_{/\mathbb {Q}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mrow> <mo stretchy="false">/</mo> <mi mathvariant="double-struck">Q</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> of rank 1 such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$E_{/K}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mrow> <mo stretchy="false">/</mo> <mi>K</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is diophantine stable at <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>. This has implications for Hilbert’s tenth problem over <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/40879_2025_872_IEq7_HTML.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="120" Type="Linedraw" Width="25" /> </InlineMediaObject> </InlineEquation>. This problem asks whether there exists an algorithm that decides in finite time whether a finite system of diophantine equations over <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/40879_2025_872_IEq8_HTML.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="120" Type="Linedraw" Width="25" /> </InlineMediaObject> </InlineEquation> has a solution.

Diophantine stability for elliptic curves on average

期刊论文
发表日期：

2025年10月23日

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Anwesh Ray,
Tom Weston

摘要

Let K be a number field and $\ell \geqslant 5$ $ℓ ⩾ 5$ a prime number. Mazur and Rubin introduced the notion of diophantine stability for a variety $X_{/K}$ $X_{/ K}$ at a prime $\ell $ $ℓ$ . We show that there is a positive density set of elliptic curves $E_{/\mathbb {Q}}$ $E_{/ Q}$ of rank 1 such that $E_{/K}$ $E_{/ K}$ is diophantine stable at $\ell $ $ℓ$ . This has implications for Hilbert’s tenth problem over . This problem asks whether there exists an algorithm that decides in finite time whether a finite system of diophantine equations over has a solution.