Constructs functorial Igusa stacks for Hodge-type Shimura varieties, yielding a sheaf on Bun_G that controls cohomology and proves compatibility with the semisimple local Langlands correspondence of Fargues-Scholze while establishing torsion vanishing for proper cases.
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abstract
We construct (cohomological) correspondences between mod $p$ fibers of different Shimura varieties and describe the fibers of these correspondences by studying irreducible components of affine Deligne-Lusztig varieties. In particular, in the case of correspondences from a Shimura set to a Shimura variety, we obtain a description of the basic Newton stratum of the latter, and show that the irreducible components of the basic Newton stratum generate all the Tate classes in the middle cohomology of the Shimura variety, under a certain genericity condition. Along the way, we also determine the set of irreducible components of the affine Deligne-Lusztig variety associated to an unramified twisted conjugacy class.
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UNVERDICTED 5representative citing papers
Orbital integrals on unitary groups over local fields in positive characteristic converge absolutely.
Proves classicality for Hecke characters in completed cohomology of Hilbert modular varieties under absolute irreducibility and regular parallel weight conditions on Galois representations, giving new cases of the LCFM conjecture.
The EKOR-stratification on the Siegel modular variety with parahoric level structure modulo p is realized as the fibers of a smooth morphism to a stack parametrizing homogeneously polarized chains of truncated displays.
Establishes arithmetic level raising for quaternionic unitary Shimura varieties of degree four at ramified primes, using supersingular locus descriptions related to Siegel threefolds.
citing papers explorer
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Igusa Stacks and the Cohomology of Shimura Varieties
Constructs functorial Igusa stacks for Hodge-type Shimura varieties, yielding a sheaf on Bun_G that controls cohomology and proves compatibility with the semisimple local Langlands correspondence of Fargues-Scholze while establishing torsion vanishing for proper cases.
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Convergence of orbital integrals on unitary groups in positive characteristic
Orbital integrals on unitary groups over local fields in positive characteristic converge absolutely.
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Classicality of Hilbert modular forms
Proves classicality for Hecke characters in completed cohomology of Hilbert modular varieties under absolute irreducibility and regular parallel weight conditions on Galois representations, giving new cases of the LCFM conjecture.
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The EKOR-stratification on the Siegel modular variety with parahoric level structure
The EKOR-stratification on the Siegel modular variety with parahoric level structure modulo p is realized as the fibers of a smooth morphism to a stack parametrizing homogeneously polarized chains of truncated displays.
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Arithmetic level raising for certain quaternionic unitary Shimura variety
Establishes arithmetic level raising for quaternionic unitary Shimura varieties of degree four at ramified primes, using supersingular locus descriptions related to Siegel threefolds.