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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 2 Pith papers
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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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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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