Signed Selmer cotorsion conjecture over the cyclotomic extension

Let EE be an elliptic curve over the number field FF', let FF be a finite extension satisfying the hypotheses of the paper, and let FcycF^{\mathrm{cyc}} be the cyclotomic Zp\mathbb{Z}_p-extension of FF, with Γ=Gal(Fcyc/F)\Gamma=\operatorname{Gal}(F^{\mathrm{cyc}}/F). For each choice s=(sv)vΣss{+,}Σss{\vec{s}}=(s_v)_{v\in\Sigma_{\mathrm{ss}}}\in\{+,-\}^{\Sigma_{\mathrm{ss}}}, let Xs(E/Fcyc)X^{\vec{s}}(E/F^{\mathrm{cyc}}) denote the Pontryagin dual of the corresponding signed Selmer group. Signed Selmer cotorsion conjecture. For all choices of s{\vec{s}}, the Zp[[Γ]]\mathbb{Z}_p[[\Gamma]]-module Xs(E/Fcyc)X^{\vec{s}}(E/F^{\mathrm{cyc}}) is torsion. For an elliptic curve over Q\mathbb{Q} with good supersingular reduction at pp, this conjecture was proved by Kobayashi; the source notes generalizations and further progress.

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Primary source

Antonio Lei and Meng Fai Lim, “Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes above p”, arXiv:2001.09304 (2020).

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