The noncommutative Selmer torsion conjecture for elliptic curves

Let E/QE/\mathbb Q be an elliptic curve, let p5p\geq5 be a prime at which EE has good ordinary reduction, and set F=Q(Ep)F_{\infty}=\mathbb Q(E_{p^{\infty}}). Let G=Gal(F/Q)G=\operatorname{Gal}(F_{\infty}/\mathbb Q) and H=Gal(F/Qcyc)H=\operatorname{Gal}(F_{\infty}/\mathbb Q^{\rm cyc}), and let MH(G)\mathfrak M_H(G) denote the category defined using the Ore set SS^*.

The noncommutative Selmer torsion conjecture. The Pontryagin dual of the Selmer group,

X(E/F)=Hom(Sel(E/F),Qp/Zp),X(E/F_{\infty})=\operatorname{Hom}(\operatorname{Sel}(E/F_{\infty}),\mathbb Q_p/\mathbb Z_p),

belongs to MH(G)\mathfrak M_H(G).

This is the basic torsion conjecture for the noncommutative extension generated by the pp-power division points of EE. The supplied text gives no resolution of it.

Sources & referencesView supporting material

Primary source

J. Coates, T. Fukaya, K. Kato, R. Sujatha and O. Venjakob, “The GL_2 main conjecture for elliptic curves without complex multiplication”, arXiv:math/0404297 (2004).

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