Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4

2023/12/18 by Kai Huang, Huang, Kai, Yongqi Liang +1
Arts and Humanities · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #North African History and Literature #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2312.11204

openalex publication_date 2023/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let g be a positive integer congruent to 1 modulo 4 and K be an arbitrary number field. We construct infinitely many explicit one-parameter algebraic families of degree 4 del Pezzo surfaces and of genus g hyperelliptic curves such that each K-member of the families violates the Hasse principle. In particular, we obtain algebraic families of non-trivial 2-torsion elements in the Tate-Shafarevich group of elliptic curves over K. These Hasse principle violations are explained by the Brauer-Manin obstruction.

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