The authors construct a canonical formal group action on μ-ordinary Igusa varieties for Hodge type Shimura data that integrates Maass-Shimura operators and, via p-adic Fourier theory, yields a unified algebra action encoding both p-adic differential operators and Hecke operators.
Hecke orbits on Shimura varieties of Hodge type
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abstract
We prove the Hecke orbit conjecture of Chai--Oort for Shimura varieties of Hodge type at odd primes of good reduction. We use a novel result for the local monodromy groups of $F$-isocrystals "coming from geometry", which refines Crew's parabolicity conjecture. In the course of the proof, we also introduce a noncommutative generalisation of Serre--Tate coordinates for formal neighbourhoods of central leaves, built upon the previous work of Caraiani--Scholze and Kim. Using these coordinates, we reinterpret Chai--Oort's notion of strongly Tate-linear subspaces and we establish upper bounds for their monodromy groups. For this step, we employ the notion of Cartier--Witt stacks, as introduced by Drinfeld and Bhatt--Lurie. Another crucial ingredient in the proof is a rigidity result proved by Chai--Oort, which shows that the relevant subspaces are strongly Tate-linear. On the way, we generalise de Jong's full faithfulness theorem for $F$-isocrystals.
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2026 2representative citing papers
Defines the convergent stack X_conv whose finitely generated quasi-coherent modules over O[1/p] are equivalent to convergent isocrystals for schemes of finite type over perfect fields, with explicit descriptions in other cases.
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$p$-adic Maass--Shimura operators on $\mu$-ordinary Igusa varieties
The authors construct a canonical formal group action on μ-ordinary Igusa varieties for Hodge type Shimura data that integrates Maass-Shimura operators and, via p-adic Fourier theory, yields a unified algebra action encoding both p-adic differential operators and Hecke operators.
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The convergent stack
Defines the convergent stack X_conv whose finitely generated quasi-coherent modules over O[1/p] are equivalent to convergent isocrystals for schemes of finite type over perfect fields, with explicit descriptions in other cases.