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• W2247234910 abstract "Let p be a prime number, and let M be a modular variety of PEL type over Fpwhich classifies abelian varieties in characteristic p with extra symmetries of a fixed PEL type.Consider the p-divisible group with extra symmetries consisting of all p-power torsions ofthe universal abelian scheme over M. The locus in M corresponding to a fixed isomorphismtype of a p-divisible group with extra symmetry is called a leaf by F. Oort. Each leaf is asmooth locally closed subvariety of the modular varietyM which is stable under all prime-to-pHecke correspondences on M. Oort conjectured that every Hecke orbit is dense in the leafcontaining it. Tools fashioned for this conjecture include (a) rigidity, (b) global monodromy,and (c) canonical coordinates. The theory of canonical coordinates generalizes the classicalSerre�Tate coordinates; it asserts that locally at the level of jet-spaces, every leaf is built upfrom p-divisible formal groups through a finite family of fibrations in a canonical way. TheHecke orbit conjecture is affirmed when M is a Siegel modular variety classifying principallypolarized abelian varieties of a fixed dimension, and also when M is a Hilbert modular varietyclassifying abelian varieties with real multiplications. The proof of the Siegel case, joint withF. Oort, uses the irreducibility of non-supersingular leaves in Hilbert modular varieties due toC.-F.Yu. That proof relies heavily on a special property of Siegel modular varieties: The set ofFp-rational points of a Siegel modular variety Ag,n is filled up by Fp-rational points of Hilbertmodular varieties contained in Ag,n. Possible directions for further progress include Tate-linearsubvarieties and p-adic monodromy. The title of this article suggests that each leaf deserves tobe viewed as a Shimura variety in characteristic p in its own right." @default.
• W2247234910 created "2016-06-24" @default.
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• W2247234910 date "2006-01-01" @default.
• W2247234910 modified "2023-09-27" @default.
• W2247234910 title "Hecke orbits as Shimura varieties in positive characteristic" @default.
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