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arXiv · 1804.00418

On the refined conjectures on Fitting ideals of Selmer groups of elliptic curves with supersingular reduction

Chan-Ho Kim·Masato Kurihara
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Abstract

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.

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Chan-Ho Kim, Masato Kurihara. 2018-04-02. On the refined conjectures on Fitting ideals of Selmer groups of elliptic curves with supersingular reduction. https://arxiv.org/abs/1804.00418

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