Conjecture A and $\mu$-invariant for Selmer groups of supersingular elliptic curves

Hamidi, Parham; Ray, Jishnu

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.1, pp. 853-886.

Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at the primes above $p$. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over ${\mathbb{Z}}_p^2$-extensions of an imaginary quadratic field where $p$ splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the $\mu$-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups, when $E$ has good supersingular reduction, which are completely analogous to the classical Selmer group, when $E$ has good ordinary reduction. In this survey paper we do not present any proofs, however, we have tried to give references of the discussed results for the interested reader.

Soit $p$ un nombre premier impair et soit $E$ une courbe elliptique sur un corps de nombres $F$ ayant bonne réduction en toute place au-dessus de $p.$ Dans cet article de synthèse, nous donnons un aperçu de certains des résultats importants sur le groupe de Selmer fin et les groupes de Selmer signés dans les tours cyclotomiques aussi que sur les groupes de Selmer signés dans les ${\mathbb{Z}}_p^2$-extensions d’un corps quadratique imaginaire, où $p$ est complètement décomposé. Nous discutons uniquement des aspects algébriques en utilisant des outils de la théorie d’Iwasawa. Nous donnons un survol de certains des résultats récents impliquant l’annulation de l’invariant $\mu$ sous l’hypothèse de la conjecture A. En outre, nous esquissons une analogie entre le groupe de Selmer classique dans le cas de bonne réduction ordinaire et le groupe de Selmer signé de Kobayashi dans le cas supersingulier. Nous mettons l’accent sur les propriétés des groupes de Selmer signés dans le cas où $E$ a bonne réduction supersingulière, qui sont complètement analogues à celles des groupes de Selmer classiques quand $E$ a bonne réduction ordinaire. Cet article ne contient pas de démonstrations, cependant pour le lecteur intéressé, nous donnons des références pour les résultats exposés dans le texte.

Classification: 11G40, 11R23, 14H52, 14G42
Keywords: Elliptic curves, Iwasawa theory, Selmer group

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