Rank gain of Jacobian varieties over finite Galois extensions

Let K be a number field, and let X→PK1 be a degree p covering branched only at 0, 1, and ∞. If K is a field containing a primitive p-th root of unity then the covering of P1 is Galois over K, and if p is congruent to 1mod6, then there is an automorphism σ of X which cyclically permutes the branch points. Under these assumptions, we show that the Jacobian of both X and X/〈σ〉 gain rank over infinitely many linearly disjoint cyclic degree p-extensions of K. We also show the existence of an infinite family of elliptic curves whose j-invariants are parametrized by a modular function on Γ0(3) and that gain rank over infinitely many cyclic degree 3-extensions of Q.