In this paper, we study the Fitting ideals of Selmer groups over finite subextensions in the cyclotomic $\mathbb{ Z}_p$-extension of $\mathbb{ Q}$ of an elliptic curve over $\mathbb{ Q}$. Especially, we present a proof of the "weak main conjecture" \`{ a} la Mazur and Tate for elliptic curves with good (supersingular) reduction at an odd prime $p$. We also prove the "strong main conjecture" suggested by the second named author under the validity of the $\pm$-main conjecture and the vanishing of a certain error term. The key idea is the explicit comparison among "finite layer objects", "$\pm$-objects", and "fine objects" in Iwasawa theory. The case of good ordinary reduction is also treated.
Categories: math.NT