ON THE STRUCTURE OF THE MORDELL-WEIL GROUPS OF THE JACOBIANS OF CURVES DEFINED BY yⁿ = f(x)

Let A be an abelian variety defined over a number field K. It is proved that for the composite field K_n of all Galois extensions over K of degree dividing n, the torsion subgroup of the Mordell-Weil group A(K_n) is finite. This is a variant of Ribet’s result ([7]) on the finiteness of torsion subgroup of A(K(ζ_∞)). It is also proved that for the Jacobians of superelliptic curves yⁿ = f(x) defined over K the Mordell-Weil group over the field generated by all nth roots of elements of K is the direct sum of a finite torsion group and a free ℤ-module of infinite rank.