# Higher composition laws and applications

### Ching-Li Chai

University of Pennsylvania, Philadelphia, United States

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## Abstract

Let p be a prime number, and let ℳ be a modular variety of PEL type over **F**p which 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 of the universal abelian scheme over ℳ. The locus in ℳ corresponding to a fixed isomorphism type of a p-divisible group with extra symmetry is called a leaf by F. Oort. Each leaf is a smooth locally closed subvariety of the modular variety ℳ which is stable under all prime-to-p Hecke correspondences on ℳ. Oort conjectured that every Hecke orbit is dense in the leaf containing it. Tools fashioned for this conjecture include (a) rigidity, (b) global monodromy, and (c) canonical coordinates. The theory of canonical coordinates generalizes the classical Serre–Tate coordinates; it asserts that locally at the level of jet-spaces, every leaf is built up from p-divisible formal groups through a finite family of fibrations in a canonical way. The Hecke orbit conjecture is affirmed when ℳ is a Siegel modular variety classifying principally polarized abelian varieties of a fixed dimension, and also when ℳ is a Hilbert modular variety classifying abelian varieties with real multiplications. The proof of the Siegel case, joint with F. Oort, uses the irreducibility of non-supersingular leaves in Hilbert modular varieties due to C.-F. Yu. That proof relies heavily on a special property of Siegel modular varieties: The set of **F**p-rational points of a Siegel modular variety **A**g,n is filled up by **F**p-rational points of Hilbert modular varieties contained in **A**g,n. Possible directions for further progress include Tate-linear subvarieties and p-adic monodromy. The title of this article suggests that each leaf deserves to be viewed as a Shimura variety in characteristic p in its own right.