Complete p-descent for Jacobians of Hermitian curves

Let X be the Fermat curve of degree q + 1 over the field k of q(2) elements , where q is some prime power. Considering the Jacobian J of X as a constan t abelian variety over the function field k(X), we calculate the multiplici ties, in subfactors of the Shafarevich-Tate group, of representations assoc iated with the action on X of a finite unitary group. J is isogenous to a p ower of a supersingular elliptic curve E, the structure of whose Shafarevic h-Tate group is also described.