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Zbl 0991.11016
Tilouine, J.; Urban, E.
Several-variable $p$-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 32, No. 4, 499-574 (1999). ISSN 0012-9593

Let $F$ be a totally real number field and $G= GSp(4)/F$. Let $P\subset G$ be a fixed parabolic subgroup. Consider the $(p,P)$-ordinary Hecke eigensystem $\lambda$, which is cohomological for a regular coefficient system. The authors show that there exists a several variable $p$-adic family $\lambda$ of $(p,P)$-nearly ordinary Hecke eigensystems containing $\lambda$ (Corollary 6.7). The corollary follows from the following two results: (i) control and freeness of the nearly ordinary part of the cohomology of the Siegel threefolds (section 6.3); (ii) finiteness and torsion freeness of the nearly ordinary cuspidal Hecke algebra over the Hida-Iwasawa algebra (section 6.4). Section 3 (control theorems for the ordinary cohomology group of ``bottom degree'') and sections 4, 5 (the complete calculation of the ordinary cohomology of the boundary of the Borel-Serre compactification) are the most technical parts of the article.
[A.Dabrowski (Szczecin)]

MSC 2000:: *11F33 Congruences for (p-adic) modular forms
11F80 Galois properties
11F46 Siegel modular groups and their modular and automorphic forms
11R23 Iwasawa theory
11G18 Arithmetic aspects of modular and Shimura varieties

Keywords: Siegel cusp eigenform; Galois representation; ordinary Hecke eigensystem; nearly ordinary Hecke eigensystems; several variable $p$-adic family; Hida-Iwasawa algebra; ordinary cohomology group; Borel-Serre compactification

Cited in: Zbl 1005.11031 Zbl 1020.11036 Zbl 1019.11502

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