In the first part, we take up the `note aded in proof' of [St] and sketch how to extend the main result to the case when A is a (non-zero) square mod l (but not a square in Q).
The second part deals with the L-series of CA. We determine the corresponding Hecke character (this is done on a fairly elementary level) and find a formula for the sign in the functional equation (or root number) of the L-series. This formula is then used to show that the order of vanishing of L(CA, s) at s = 1 and the Mordell-Weil rank of the Jacobian JA of CA (over Q) have the same parity (under the assumption that Sha(Q, JA) has finite l-part) for those A that are covered by the (extended) main result of [St].
Reference: [St] M. Stoll: On the arithmetic of the curves y2 = xl + A and their Jacobians, J. reine angew. Math. 501, 171-189 (1998).