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<p>Let <i>k</i> be a number field and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\pi :X \rightarrow \mathbb {P}_k^1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <msubsup> <mi mathvariant="double-struck">P</mi> <mi>k</mi> <mn>1</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> be a smooth conic bundle. We show that if <i>X</i>/<i>k</i> has four geometric singular fibers and either <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(X(\mathbb {A}_k)\ne \emptyset \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">A</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mrow> </math></EquationSource> </InlineEquation> or <i>X</i>/<i>k</i> has non-trivial Brauer group, then <i>X</i> satisfies the Hasse principle over any even degree extension <i>L</i>/<i>k</i>. Furthermore for arbitrary <i>X</i> we show that, conditional on Schinzel’s hypothesis, <i>X</i> satisfies the Hasse principle over all but finitely many quadratic extensions of <i>k</i>. We prove these results by showing the Brauer-Manin obstruction vanishes and then apply fibration method results of Colliot-Thélène, following Colliot-Thélène and Sansuc.</p>

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On the Hasse Principle for Conic Bundles over Even Degree Extensions

  • 期刊论文

  • 发表日期：

    2026年3月23日

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  • Sam Roven,
  • Alexander Wang

摘要

Let k be a number field and let \(\pi :X \rightarrow \mathbb {P}_k^1\) π : X → P k 1 be a smooth conic bundle. We show that if X/k has four geometric singular fibers and either \(X(\mathbb {A}_k)\ne \emptyset \) X ( A k ) ≠ ∅ or X/k has non-trivial Brauer group, then X satisfies the Hasse principle over any even degree extension L/k. Furthermore for arbitrary X we show that, conditional on Schinzel’s hypothesis, X satisfies the Hasse principle over all but finitely many quadratic extensions of k. We prove these results by showing the Brauer-Manin obstruction vanishes and then apply fibration method results of Colliot-Thélène, following Colliot-Thélène and Sansuc.