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We explain how to use results from Iwasawa theory to obtain information about p-parts of Tate-Shafarevich groups of specific elliptic curves over Q. Our method provides a practical way to compute #X(E/Q)(p) in many cases when traditional p-descent methods are completely impractical and also in situations where results of Kolyvagin do not apply, e.g., when… (More)

Let E/Q be an elliptic curve. We investigate the denominator of the modular symbols attached to E. We show that one can change the curve in its isogeny class to make these denominators coprime to any given odd prime of semi-stable reduction. This has applications to the integrality of Kato's Euler system and the main conjecture in Iwasawa theory for… (More)

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