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arXiv · 2312.11204

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

Kai Huang·Yongqi Liang
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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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Kai Huang, Yongqi Liang. 2023-12-18. Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4. https://arxiv.org/abs/2312.11204

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