  Let E/Q be an elliptic curve and p be a prime number, and let G be the Galois
group of the extension of Q obtained by adjoining the coordinates of the
p-torsion points on E. We determine all cases when the Galois cohomology group
H^1(G, E[p]) does not vanish, and investigate the analogous question for E[p^i]
when i>1. We include an application to the verification of certain cases of the
Birch and Swinnerton-Dyer conjecture, and another application to the
Grunwald-Wang problem for elliptic curves.
