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<p>We introduce a new quadratic Chabauty method to compute the integral points on certain even degree hyperelliptic curves. Our approach relies on a nontrivial degree zero divisor supported at the two points at infinity to restrict the <i>p</i>-adic height to a linear function; we can then express this restriction in terms of holomorphic Coleman integrals under the standard quadratic Chabauty assumption. Then we use this linear relation to extract the integral points on the curve. We also generalize our method to integral points over number fields. Our method is significantly simpler and faster than all other existing versions of the quadratic Chabauty method. We give examples over ℚ and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb{Q(\sqrt{7})}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">Q</mi> <mo mathvariant="double-struck" stretchy="false">(</mo> <msqrt> <mn mathvariant="double-struck">7</mn> </msqrt> <mo mathvariant="double-struck" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Linear quadratic Chabauty

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月29日

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  • Stevan Gajović,
  • J. Steffen Müller

摘要

We introduce a new quadratic Chabauty method to compute the integral points on certain even degree hyperelliptic curves. Our approach relies on a nontrivial degree zero divisor supported at the two points at infinity to restrict the p-adic height to a linear function; we can then express this restriction in terms of holomorphic Coleman integrals under the standard quadratic Chabauty assumption. Then we use this linear relation to extract the integral points on the curve. We also generalize our method to integral points over number fields. Our method is significantly simpler and faster than all other existing versions of the quadratic Chabauty method. We give examples over ℚ and \(\mathbb{Q(\sqrt{7})}\) Q ( 7 ) .