Generalized fine Selmer torsion conjecture over pp-adic Lie extensions

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Let Aˉ\bar{A} be one of the coefficient objects considered in the source, with coefficient ring Rˉ\bar{R}, and let F∞/FF_\infty/F be a pp-adic Lie extension such that G=Gal⁡(F∞/F)G=\operatorname{Gal}(F_\infty/F) is pro-pp, has no pp-torsion, and F∞/FF_\infty/F is unramified outside a finite set of primes. Let Y(Aˉ/F∞)Y(\bar{A}/F_\infty) be the Pontryagin dual of the corresponding fine Selmer group. Generalized fine Selmer torsion conjecture. For every such pp-adic Lie extension,

Y(Aˉ/F∞) is torsion over Rˉ[[G]].Y(\bar{A}/F_\infty)\text{ is torsion over }\bar{R}[[G]].

This extends the Zp\mathbb{Z}_{p}-extension torsion conjectures to noncommutative pp-adic Lie extensions and is motivated by known torsion results for related étale wild kernels. The generalized assertion remains open in the stated generality.

References

Primary source

Meng Fai Lim, “Structure of fine Selmer groups over Z_p-extensions”, arXiv:2111.08866 (2023).

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