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. Let E be an elliptic curve over Q and p be a prime of goodsupersingular reduction for E. Although the Iwasawa theory of Eover thecyclotomic Z p -extension of Qis well known to be fundamentally dierentfrom the case of good ordinary reduction at p, we are able to combine themethod of our earlier paper with the theory of Kobayashi [5] and Pollack[8], to give an explicit upper bound for the number of copies of Q p /Z p occurring in the p-primary part of the Tate-Shafarevich group of E overQ. 1. IntroductionLet E be an elliptic curve over Q. We recall that the Tate-Shafarevich groupof E/Qis dened byX(E/Q) = Ker H 1 Q,E(Q)−→Y v H 1 Q v ,E(Q )!,where v runs over all places of Q, and Q v is the completion of Q at v. It iswell-known that the p-primary subgroup of X(E/Q) has a nite Z p -corank,and we denote this corank by t p . It is conjectured that t p = 0 for every primep, but this is unknown when the complex L-function has a zero of order atleast 2 at s = 1. We say a prime p is good ordinary (resp. good supersingular)if E has good ordinary reduction (resp. good supersingular reduction) at p.The aim of present paper is to compute an upper bound for t

ON THE p-PRIMARY PART OF TATE-SHAFAREVICH GROUP OF ELLIPTIC CURVES OVER ℚ WHEN p IS SUPERSINGULAR