Schneider's rank conjecture for Selmer groups over p-adic Lie extensions

From papers

Let AA be an abelian variety over a finite extension FF of Q\mathbb Q, let p>2dim(A)+1p>2\dim(A)+1 be prime, let F=F(Ap)F_\infty=F(A_{p^\infty}), and put Σ=Gal(F/F)\Sigma=\operatorname{Gal}(F_\infty/F). Let RR be the relevant closed subgroup of Σ\Sigma, let EE be an elliptic curve over FF, let Λ(R)\Lambda(R) be the Iwasawa algebra of RR, and let τp(E/F)\tau_p(E/F) denote the quantity defined in the source. Write C(E/F)\mathcal C(E/F_\infty) for the Pontryagin dual of Sel(E/F)\operatorname{Sel}(E/F_\infty). Schneider's rank conjecture. One has

rkΛ(R)C(E/F)=[Σ:R]τp(E/F).\operatorname{rk}_{\Lambda(R)}\mathcal C(E/F_\infty)=[\Sigma:R]\tau_p(E/F).

The equality is presented as a natural generalisation of a conjecture in Schneider. The preceding corollary gives only the corresponding upper and lower bounds, so the asserted equality remains open in the source.

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Sources & referencesView supporting material

Primary source

Sarah Livia Zerbes, “Selmer Groups over p-adic Lie Extensions I”, arXiv:math/0404203 (2004).

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