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Several-variable \(p\)-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations. (English) Zbl 0991.11016
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.

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11F80 Galois representations
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11R23 Iwasawa theory
11G18 Arithmetic aspects of modular and Shimura varieties
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