On the variation of Tate–Shafarevich groups of elliptic curves over hyperelliptic curves Original Research Article

Let E be an elliptic curve over image having conductor image, where image is a prime ideal in image. Let image be an irreducible polynomial of odd degree, and let image. Assume image remains prime in K. We prove the analogue of the formula of Gross for the special value L(Ecircle times operatorFK,1). As a consequence, we obtain a formula for the order of the Tate–Shafarevich group Ш(E/K) when L(Ecircle times operatorFK,1)≠0.