Curves over global fields violating the Hasse Principle

TL;DR

This paper constructs explicit examples of algebraic curves over global fields that violate the Hasse Principle, using refined twisting techniques and modular curves, addressing a question posed by B. Poonen.

Contribution

It provides the first known examples of curves over all global fields that violate the Hasse Principle, employing a refined Twist Anti-Hasse Principle method.

Findings

Explicit curves violating the Hasse Principle over all global fields.

Construction of such curves among Atkin-Lehner twists of modular curves.

Extension of the method to curves of any genus ≥ 2.

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.