Control theorems of coherent sheaves on Shimura varieties of PEL type. (English) Zbl 1039.11041

Given a reductive algebraic group \(G\) defined over \(\mathbb Q\), a Shimura variety \(\text{Sh}_K\) can be constructed by using a compact open subgroup \(K\) of \(G (\mathbb A)\). This paper deals with coherent sheaves on Shimura varieties which carry canonical families of abelian varieties with PEL structure. The groups producing such Shimura varieties include inner forms of GSp\((2g)\) over totally real fields and inner forms of unitary groups GU\((m,n)\) relative to a CM field. As is well known, each Shimura variety is defined over a reflex field inside the algebraic closure \(\overline{\mathbb Q}\) of \(\mathbb Q\) in \(\mathbb C\). Given a prime \(p\), if \(K\) is of level prime to \(p\), the associated Shimura variety \(\text{Sh}_K\) extends canonically to a scheme faithfully flat over a \(p\)-adic valuation ring of the reflex field.
In this paper the author proves an exact control theorem for nearly ordinary \(p\)-adic automorphic forms on symplectic and unitary groups over totally real fields when the algebraic group is split over \(p\). In particular, he shows that 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 groups. 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 certain assumptions on the existence of integral toroidal compactification of the variety.


11G18 Arithmetic aspects of modular and Shimura varieties
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F33 Congruences for modular and \(p\)-adic modular forms
11F30 Fourier coefficients of automorphic forms
14G35 Modular and Shimura varieties
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