  This paper introduces explicit Galois cohomological methods for determining
the ranks of Bloch--Kato Selmer groups associated to the Tate twists of the
2-adic second \'etale cohomology of the Jacobian of a hyperelliptic curve with
a rational Weierstrass point. In particular, this can give a method to
determine the rational points on such curves via the Chabauty--Coleman--Kim
method. This is applied to answer a question of Bugeaud, Mignotte, Siksek,
Stoll and Tengely.
