The finite-level ell-independence conjecture for unipotent fundamental groups

Let X/FX/F be a geometrically connected variety with base point xX(F)x\in X(F). For each 0˘eell\u0**eell, let U^0˘eell,X,x\hat{\mathcal{U}}_{\u0**eell,X,x} be the completed universal enveloping algebra of the Lie algebra of the ell-adic unipotent fundamental group, and let a0˘eell,X,x\mathfrak{a}_{\u0**eell,X,x} be its augmentation ideal. Fix k1k\geq 1. Finite-level unipotent ell-independence conjecture. The system {U^0˘eell,X,x/a0˘eell,X,xk}0˘eell\{\hat{\mathcal{U}}_{\u0**eell,X,x}/\mathfrak{a}_{\u0**eell,X,x}^k\}_{\u0**eell} of WFW_F'-representations is (weakly) Q{\mathbb Q}-compatible. This is presented as a weakening of the conjecture for the full unipotent fundamental group; the source does not indicate that it has been resolved.

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Primary source

Bruno Chiarellotto and Christopher Lazda, “Around -independence”, arXiv:1608.03796 (2017).

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