Models of some genus one curves with applications to descent

Let p denote a prime, and K a field of characteristic prime to p and containing the pth roots of unity. For p equal to 3 and 5, the author finds a scheme Tp and a family of genus one curves over Tp such that any genus one curve defined over the field K of index p whose Jacobian elliptic curve E has E[p](K)=E[p](K¯) is isomorphic to a curve lying over a K-point of Tp. The author then relates the explicit presentation of such families to the program of descent on elliptic curves.