*p*-Adic Automorphic Forms on Shimura Varieties
pp 329-374 |
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# Ordinary *p*-Adic Automorphic Forms

## Abstract

We first describe a formal theory of false automorphic forms and find a set of conditions for nearly ordinary *p*-adic cusp forms to have vertical control. After this, we describe deformation theory by Serre-Tate of abelian varieties in order to prove the divisibility by the character value *μ*(ξ) in (5.40) of the Hecke operator associated with an element ξ of the expanding semi-group *D* in Section 5.1.2. By this, we have well-defined *p*-integral Hecke operators *T*_{ p }(ξ) on coherent cohomology groups and therefore well-defined nearly *p*-ordinary automorphic forms. Verifying the set of conditions on sections of automorphic vector bundles of a given (projective) Shimura variety of PEL type, we prove the vertical control theorem for automorphic forms on the Shimura varieties. We also state a similar result for quasi-projective Shimura varieties of quasi-split unitary groups without a detailed proof. At the end, we prove the irreducibility of the Igusa tower over the ordinary locus of symplectic and unitary Shimura varieties, reducing the problem to the Siegel modular case. By means of the Serre-Tate deformation coordinates, even when the Shimura variety does not have a cusp, we are able to define an analogue of the *p*-adic valuation *v* (in (4.36)) of its arithmetic automorphic function field; so, the irreducibility can be proven even for projective symplectic and unitary Shimura varieties of PEL type.

## Keywords

Irreducible Component Parabolic Subgroup Abelian Variety Automorphic Form Maximal Parabolic Subgroup## Preview

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