Algebraic families of hyperelliptic curves violating the Hasse principle

TL;DR

This paper generalizes previous work on genus one curves to construct algebraic families of high-genus hyperelliptic curves that violate the Hasse principle, explained by the Brauer-Manin obstruction.

Contribution

It extends the construction of Hasse principle violations from genus one to arbitrary high-genus hyperelliptic curves for specific genera.

Findings

Existence of algebraic families of hyperelliptic curves of genus n violating the Hasse principle.

Violations are explained by the Brauer-Manin obstruction.

Results apply for all n > 5, n ≡ 1, 2, or 3 mod 4.

Abstract

In $2000$, Colliot-Th\'el\`ene and Poonen showed how to construct algebraic families of genus one curves violating the Hasse principle. Poonen explicitly constructed an algebraic family of genus one cubic curves violating the Hasse principle using the general method developed by Colliot-Th\'el\`ene and himself. The main result in this paper generalizes the result of Colliot-Th\'el\`ene and Poonen to arbitrarily high genus hyperelliptic curves. More precisely, for $n > 5$ and $n \not\equiv 0 \pmod{4}$, we show that there is an algebraic family of hyperelliptic curves of genus $n$ that is counterexamples to the Hasse principle explained by the Brauer-Manin obstruction.