Computations on the Birch and Swinnerton-Dyer conjecture for elliptic curves over pure cubic extensions

Computations on the Birch and Swinnerton-Dyer conjecture for elliptic curves over pure cubic extensions Céline MaistretThe Birch and Swinnerton-Dyer conjecture remains an open problem.In this thesis, we propose to give numerical evidence toward this conjecture when restricted to elliptic curves over pure cubic extensions.We present the general conjecture for elliptic curves over number fields and detail each arithmetic invariants involved.Assuming the conjecture holds, for given elliptic curves E over specific number fields K, we compute the order of the Shafarevich-Tate group of E(K).iii