The non-abelian zeta-distribution main conjecture for Artin motives

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Let MM be an Artin motive, let k>1k>1, let SS and G∞G_\infty be as before with G∞G_\infty pro-pp and without pp-torsion, and assume that Q[Gn]⊗M(k){\Bbb Q}[G_n]\otimes M(k) is critical for all nn. Let Λ=Zp[[G∞]]\Lambda={\Bbb Z}_p[[G_\infty]] and let DD be the total ring of fractions used in the paper. The zeta distribution is

LS(G∞,M∨,1−k)=lim←⁡LS(Gn,M∨,1−k)∈lim←⁡Z(Qp[Gn])∗.\mathcal{L}_S(G_\infty,M^\lor,1-k)=\varprojlim L_S(G_n,M^\lor,1-k)\in\varprojlim Z({\Bbb Q}_p[G_n])^*.

Zeta-distribution main conjecture. There exist f,g∈Λf,g\in\Lambda such that (rn⁡(fngn−1))n(\operatorname{rn}(f_ng_n^{-1}))_n is the zeta distribution LS(G∞,M∨,1−k)\mathcal{L}_S(G_\infty,M^\lor,1-k) and the characteristic ideal

[RΓ(Z[1/S],Λ⊗Tp(k))[1]]∈K0(T)[R\Gamma({\Bbb Z}[1/S],\Lambda\otimes T_p(k))[1]]\in K_0(\mathcal{T})

coincides with the image of fg−1∈(D∗)abfg^{-1}\in(D^*)^{\mathrm{ab}}.

This is the torsion, critical Artin-motive specialization of the non-abelian Iwasawa Main Conjecture, relating zeta distributions to characteristic ideals. The source gives no resolution evidence.

References

Primary source

Annette Huber and Guido Kings, “Equivariant Bloch-Kato conjecture and non-abelian Iwasawa Main Conjecture”, arXiv:math/0207284 (2002).

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