In 2000, Colliot-Thélène and Poonen showed how to construct algebraic families of
genus-one curves violating the Hasse principle. Poonen explicitly constructed such a
family of cubic curves using the general method developed by Colliot-Thélène and
himself. The main result in this paper generalizes the result of Colliot-Thélène and
Poonen to arbitrarily high genus hyperelliptic curves. More precisely, for
and
, we
show that there is an explicit algebraic family of hyperelliptic curves of genus
that
are counterexamples to the Hasse principle explained by the Brauer–Manin
obstruction.
