Explicit construction of decomposable Jacobians

In this note we give explicit constructions of decomposable hyperelliptic Jacobian varieties over fields of characteristic $0$. These include hyperelliptic Jacobian varieties that are isogenous to a product of two absolutely simple hyperelliptic Jacobian varieties, a square of a hyperelliptic Jacobian variety, and a product of four hyperelliptic Jacobian varieties three of which are of the same dimension. As an application, we produce families of hyperelliptic curves with infinitely many quadratic twists having at least two rational non-Weierstrass points; and families of quadruples of hyperelliptic curves together with infinitely many square-free $d$ such that the quadratic twists of each of the curves by $d$ possess at least one rational non-Weierstrass point.