Algebraic families of hyperelliptic curves violating the Hasse principle

Dong Quan Ngoc Nguyen

Published 2014-07-27Version 1

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.