Tate-Shafarevich groups of Jacobians of Fermat curves
For a fixed rational prime p and primitive p-th root of unity (, we consider the Jacobian, J, of the complete non-singular curve give by equation yP = xa(1 - x)b. These curves are quotients of the p-th Fermat curve, given by equation xP+yP = 1, by a cyclic group of automorphisms. Let k = Q(() and ks be the maximal extension of k unramified away from p inside a fixed algebraic closure of k. We produce a formula for the image of certain coboundary maps in group cohomology given in terms of Massey products, applicable in a general setting. Under specific circumstance, stated precisely below, we can use this formula and a pairing in the Galois cohomology of ks over k studied by W. McCallum and R. Sharifi in [MS02] to produce non-trivial elements in the Tate-Shafarevich group of J. In particular, we prove a theorem for predicting when the image of certain cyclotomic p-units in the Selmer group map non-trivially into X(k, J).