The intersection-cohomology ordinariness conjecture

Let XX be a projective variety over a number field FF, let nn be an integer, and let FsF^s be a separable closure of FF. For every prime number \ell, consider the Newton polygon NP(FrobpIHn(XFFs,Q))\operatorname{NP}(\operatorname{Frob}_{\mathfrak{p}}|_{IH^n(X\otimes_FF^s,\mathbb{Q}_{\ell})}) and the Hodge–Tate polygon HTP(IHn(XFFs,Q))\operatorname{HTP}(IH^n(X\otimes_FF^s,\mathbb{Q}_{\ell})).

Intersection-cohomology ordinariness conjecture. For every prime number \ell, there exists an infinite set of maximal ideals p\mathfrak{p} of OF\mathscr{O}_F with residue characteristic different from \ell such that

NP(FrobpIHn(XFFs,Q))=HTP(IHn(XFFs,Q)).\operatorname{NP}(\operatorname{Frob}_{\mathfrak{p}}|_{IH^n(X\otimes_FF^s,\mathbb{Q}_{\ell})})=\operatorname{HTP}(IH^n(X\otimes_FF^s,\mathbb{Q}_{\ell})).

This is presented as the analogue of the ordinariness conjecture for intersection cohomology. The source provides partial verification in special motivic settings but no general resolution.

Sources & referencesView supporting material

Primary source

Junecue Suh, “Ordinary primes in Hilbert modular varieties”, arXiv:2410.01182 (2024).

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