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Higher Coleman Theory
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We develop local cohomology techniques to study the finite slope part of the coherent cohomology of Shimura varieties. The local cohomology groups we consider are a generalization of overconvergent modular forms, and they are defined by using a stratification on the Shimura variety obtained from the Bruhat stratification on a flag variety via the Hodge-Tate period map. We construct a spectral sequence from the local cohomologies to the classical cohomology and use it to obtain classicality and vanishing results. We also develop a theory of p-adic families and construct eigenvarieties. As an application, we prove some new properties of Galois representations arising from certain non-regular algebraic cuspidal automorphic representations.
Forward citations
Cited by 4 Pith papers
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Locally analytic completed cohomology of Shimura varieties of Hodge type
For Shimura varieties of Hodge type, locally analytic completed cohomology is naturally isomorphic to cohomology of flag varieties with locally analytic sheaves from perfectoid covers.
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Overconvergent Eichler-Shimura morphisms for $\mathrm{GSp}_4$
The full four-step Eichler-Shimura decomposition for GSp4 is interpolated along p-adic weight families, and near any nice-enough eigenvariety point it splits with Hodge-Tate-Sen weights (-3, κ2-2, κ1-1, κ1+κ2).
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$p$-adic Asai and twisted triple product $L$-functions for finite slope families
The paper constructs two-variable p-adic Asai and twisted triple product L-functions for finite-slope families of Hilbert modular forms over real quadratic fields.
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New perspectives on $p$-adic regulator formulae
A finite-slope p-adic regulator formula for Asai–Flach classes is proved without finite-polynomial cohomology, and diagonal classes are reconstructed as pullback extension classes.
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