Journal of the Institute of Mathematics of Jussieu



CONTROL THEOREMS OF COHERENT SHEAVES ON SHIMURA VARIETIES OF PEL TYPE


Haruzo Hida a1
a1 Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA

Abstract

An exact control theorem is proven for nearly ordinary $p$-adic automorphic forms on symplectic and unitary groups over totally real fields if the algebraic group is split at $p$. In particular, a given nearly ordinary holomorphic Hecke eigenform can be lifted to a family of holomorphic Hecke eigenforms indexed by weights of the standard maximal split torus of the group. Their $q$-expansion coefficients are Iwasawa functions on the Iwasawa algebra of ${\mathbb{Z}}_p$-points of the split torus. The method is applicable to any reductive algebraic groups yielding Shimura varieties of PEL type under mild assumptions on the existence of integral toroidal compactification of the variety. Even in the Hilbert modular case, the result contains something new: freeness of the universal nearly ordinary Hecke algebra over the Iwasawa algebra, which eluded my scrutiny when I studied general theory of the $p$-adic Hecke algebra in the 1980s.

AMS 2000 Mathematics subject classification: Primary 11F03; 11F30; 11F33; 11F41; 11F60; 11G15; 11G18

(Received June 14 2000)
(Accepted January 9 2001)


Key Words: Hecke algebra; Hecke operator; $p$-adic automorphic forms; $p$-adic family.