Mathematical Research Letters 3, 309–318 (1996) EFFECTIVE BOUNDS ON THE SIZE OF THE TATE-SHAFAREVICH GROUP Dorian Goldfeld and Daniel Lieman Throughout this paper we will work over Q, although our methods gen- eralize quite naturally to any number field. Consider an elliptic curve E defined over Q. We use the following nota- tion: X E denotes the Tate-Shafarevich group of E, N E denotes the conduc- tor of E, j(E) denotes the j-invariant of E, Δ(E) denotes the discriminant of E, and |S| denotes the cardinality of any set S . It has been conjectured by Goldfeld and Szpiro [GS] that for every ǫ> 0 there exists an effectively computable constant c> 0 depending only on ǫ such that |X E | <cN 1 2 +ǫ . Goldfeld and Szpiro also show that if this conjecture holds for rank zero semistable elliptic curves, a version of the ABC conjecture follows. In particular, for coprime integers A,B and C satisfying A + B + C = 0 one has |ABC | 1/3 = O    p | ABC p   3+2ǫ ; if in addition one assumes a Lindel¨ of hypothesis, one may improve this to |ABC | 1/3 = O    p | ABC p   1+ 13 6 ǫ . We prove the conjecture of Goldfeld and Szpiro (subject to various stan- dard conjectures) for any collection C of elliptic curves which has the prop- erties that: 1) the set {j(E) | E ∈ C} is finite; and 2) C does not contain any curves with j-invariant 0 or 1728. In many cases, the conjectures we assume Received November 30, 1995. Both authors are partially supported by the National Science Foundation. 309