On the Twist of Abelian Varieties Defined by the Galois Extension of Prime Degree Original Research Article

Let p be a prime number and let k be a field which contains a primitive pth root of unity. For the curve C over k defined by up = ƒ(t), and an abelian variety A over k which has a complex multiplication by Z[ω], where ω = exp(2πi/p), the Mordell-Weil group of the twist of A defined by the extension k(C)/k(P1) is isomorphic with the direct sum of a subgroup of Homk(J(C), A) (J(C) is the jacobian variety of C) and the group of k-rational (1 − ω)-division points of A.