Torsion points of elliptic curves over multi-quadratic number fields

We compute the Mordell–Weil groups of the modular Jacobian varieties of hyperelliptic modular curves X1(M,MN) over every composite field of some quadratic number fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman.