On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=Fq(t) having conductor (p)·∞, where (p) is a prime ideal in Fq[t]. Let d∈Fq[t] be an irreducible polynomial of odd degree, and let K=F(d). Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗FK,1)≠0.