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The Shafarevich-Tate and Selmer groups arise in the context of Kummer theoryfor elliptic curves. The finiteness of the Shafarevich-Tate group of anelliptic curve $E$ over the field of rational numbers is included in the Birchand Swinnerton-Dyer conjectures, and is still an open question.  We present an overview of the Shafarevich-Tate and Selmer groups of anelliptic curve in the framework of Galois cohomology. Known results on thefiniteness of the Shafarevich-Tate group are mentioned, including results ofCoates and Wiles, Rubin, Gross and Zagier, and Kolyvagin.  We then prove the vanishing of the $\ell$-primary component of theShafarevich-Tate group for almost all primes $\ell$, for any elliptic curve $E$over the rationals without complex multiplication.

On the vanishing of almost all primary components of the Shafarevich-Tate group of elliptic curves over the rationals