THE HASSE PRINCIPLE FOR CERTAIN HYPERELLIPTIC CURVES AND FORMS

We prove that, for each positive integer n >= 2, there is an infinite arithmetic family of hyperelliptic curves of genus n violating the Hasse principle explained by the Brauer-Manin obstruction. Using these families of curves, we show that, for any positive integer k >= 1, there are infinitely many algebraic and arithmetic families of forms in three variables of degree 4k+2 such that they are counterexamples to the Hasse principle explained by the Brauer-Manin obstruction.