Curves over global fields violating the Hasse Principle

2009/05/21 by Pete L. Clark, Clark, Pete L.
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0905.3459

openalex publication_date 2009/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In response to a question of B. Poonen, we exhibit for each global field k an algebraic curve over k which violates the Hasse Principle. In fact we can find such examples among Atkin-Lehner twists of certain elliptic modular curves and -- in positive characteristic -- Drinfeld modular curves. Our main tool is a refinement of the "Twist Anti-Hasse Principle" (TAHP). We also use TAHP to construct further Hasse Principle violations, for instance among curves over any number field of any given genus g which is at least 2.

Citations

• The period-index problem in WC-groups II: abelian varieties

Related

• An "Anti-Hasse Principle" for Prime Twists
• Hasse principle violations in twist families of superelliptic curves
• Algebraic families of hyperelliptic curves violating the Hasse principle
• On the Hasse principle for Shimura curves
• Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4