Scholarlink

<p>Using the recent work of S. Y. Xiao [Prime values of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f\left( a, b^2\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mfenced close=")" open="("> <mi>a</mi> <mo>,</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(f\left( a, p^2\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mfenced close=")" open="("> <mi>a</mi> <mo>,</mo> <msup> <mi>p</mi> <mn>2</mn> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, <i>f</i> quadratic, Algebra Number Theory, 18 (9) (2024), 1619–1679], we construct an infinite family of degree 4 del Pezzo surfaces that violate the Hasse principle. This family contains the surface studied by Birch and Swinnerton-Dyer [The Hasse problem for rational surfaces, J. Reine Angew. Math. 274/275 (1975) 164–174] and the family studied by Quan [The arithmetic of certain del Pezzo surfaces and <i>K</i>3 surfaces, J. Théor. Nombres Bordx. 24 (2) (2012) 447–460].</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

An infinite family of del Pezzo surfaces violating the Hasse principle

  • 期刊论文

  • 发表日期：

    2025年11月15日

查看全文
  • Nguyen Xuan Tho,
  • Duc Hiep Pham

摘要

Using the recent work of S. Y. Xiao [Prime values of \(f\left( a, b^2\right) \) f a , b 2 and \(f\left( a, p^2\right) \) f a , p 2 , f quadratic, Algebra Number Theory, 18 (9) (2024), 1619–1679], we construct an infinite family of degree 4 del Pezzo surfaces that violate the Hasse principle. This family contains the surface studied by Birch and Swinnerton-Dyer [The Hasse problem for rational surfaces, J. Reine Angew. Math. 274/275 (1975) 164–174] and the family studied by Quan [The arithmetic of certain del Pezzo surfaces and K3 surfaces, J. Théor. Nombres Bordx. 24 (2) (2012) 447–460].