Non-abelian Leopoldt conjecture for ordinary completed cohomology

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Let FF be a number field, pp a prime, nn a positive integer, U⊂GL⁡n(AF∞)U\subset\operatorname{GL}_n({\mathbb A}_F^\infty) a good subgroup, and m⊂TordS(U){\mathfrak m}\subset{\mathbb T}^{S}_\text{ord}(U) a non-Eisenstein maximal ideal. Non-abelian Leopoldt conjecture.

dim⁡Λ[1/p]Hord∗(U)m[1/p]=dim⁡Λ[1/p]−l0.\dim_{\Lambda[1/p]}H^\ast_\text{ord}(U)_{\mathfrak m}[1/p]=\dim\Lambda[1/p]-l_0.

This is the non-abelian analogue of Leopoldt's conjecture, phrased as a dimension formula for localized ordinary completed cohomology over the weight Iwasawa algebra. The source gives no resolution status.

References

Primary source

Chandrashekhar Khare and Jack A. Thorne, “Potential automorphy and the Leopoldt conjecture”, arXiv:1409.7007 (2016).

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