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

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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

rk⁡Λ(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.

References

Primary source

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

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