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- W4386869604 abstract "As seen in Chap. 8 , originally proved with Michael Groechenig in Esnault and Groechenig (Acta Math 225(1):103–158, 2020, Theorem 1.6), rigid connections on $$X_{{mathbb C}}$$ smooth projective over $${mathbb C}$$ , while restricted to the formal p-completion $$hat X_W$$ a non-ramified projective p-adic model $$X_W$$ of $$X_{mathbb C}$$ , yield F-isocrystals. More is true. By showing in Esnault and Groechenig (Acta Math 225(1):103–158, 2020, Theorem 1.6) the existence of a periodic Higgs-de Rham flow on the formal connection $$(hat E_W, hat nabla _W)$$ on $$hat X_W$$ , we prove the existence of a Fontaine-Lafaille module structure on $$(hat E_W, hat nabla _W)$$ (Esnault and Groechenig, Acta Math 225(1):103–158, 2020, Theorem 1.6, Section 4), which, via Faltings’ functor, eventually yields a crystalline $${mathbb Z}_{p^f}$$ -local system on the algebraic scheme $$X_K$$ , where f is the period of the Higgs-de Rham flow. This in turn implies that the rigid complex local systems on $$X_{mathbb C}$$ , for $$p>0$$ large so they are integral by p, the residual characteristic of such a good W, descend to crystalline $${mathbb Z}_{p^f}$$ -local systems, see Esnault and Groechenig (Acta Math 225(1):103–158, 2020, Section 5). This property remains true even if X is only quasi-projective under a strong cohomological rigidity assumption, which is fulfilled on Shimura varieties of real rank $$ge 2$$ , and assuming in addition that the local monodromies at infinity are unipotent, see Esnault and Groechenig (Frobenius structures and unipotent monodromy at infinity, Preprint 2021, 8 pp., Theorem A.4, Theorem A.22)." @default.
- W4386869604 created "2023-09-20" @default.
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- W4386869604 date "2023-01-01" @default.
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- W4386869604 title "Lecture 9: Rigid Local Systems, Fontaine-Laffaille Modules and Crystalline Local Systems" @default.
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- W4386869604 doi "https://doi.org/10.1007/978-3-031-40840-3_9" @default.
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