Tate-Shafarevich Groups of Jacobians of Fermat Curves

Abstract

For a fixed rational prime p and primitive p-th root of unity ζ, 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(ζ) and k(S) 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 k(S) 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).

Q(zeta) and k_S 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 k_S over k studied by W. McCallum and R. Sharifi 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 Shah(k,J).