Development of an algorithm for the calculation of the Tate-Shafarevich groups and applicaiton

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We develop an algorithm for the calculation of the Tate-Shafarevich groups of elliptic curves defined over number fields. We also make a numerical verification of a conjecture by Greenberg for 2-adic Iwasawa mu-invariants of elliptic curves and 2-adic Iwasawa main conjecture for elliptic curves by using a relation between mu-invariants of elliptic curves and partly ramified Iwasawa modules associated with certain number fields. Moreoer, we find some examples related to the problem about the rank of the Mordell-Weil group of elliptic curves over number fields.