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\(p\)-adic families of Siegel modular cuspforms. (English) Zbl 1394.11045

The main result of this important paper is a geometric constructions of the Siegel eingenvariety and \(p\)-adic families of Siegel cuspforms.
Eigenvarieties and \(p\)-adic families of modular forms have been defined and studied with different approaches. One can study the cohomology of towers of Shimura varieties with \(p\)-power levels and trivial coefficients (this is the approach of Hida, Emerton and others), or one can fix a Shimura variety (with level structure of the form \(\Gamma_0(Np)\) with \((N,p)=1\)) and study its cohomology with coefficients in more complicate spaces of locally analytic distributions (this is the approach of Stevens, Urban and others). The approach of Coleman and Hida to the construction of the eigencurve was based on Katz’s theory of \(p\)-adic and overconvergent modular forms: the idea is to interpolate directly classical modular forms by overconvergent modular forms, which are allowed to have singularities in small \(p\)-adic disks centred at supersingular points (called supersingular disks). In the case of higher rank groups, this geometric approach only works for ordinary families or one-dimensional finite slope families.
In previous papers, F. Andreatta et al. [Isr. J. Math. 201, Part A, 299–359 (2014; Zbl 1326.14051)] and V. Pilloni [Ann. Inst. Fourier 63, No. 1, 219–239 (2013; Zbl 1316.11034)] proposed (independently) a new geometric approach to the construction of the eigencurve and \(p\)-adic families of (elliptic) modular forms. The basic idea is to interpolate over the complement of supersingular disks the automorphic sheaves themselves, and construct \(p\)-adic families of such automorphic sheaves. Then Coleman \(p\)-adic families of modular forms are just global sections of a \(p\)-adic family of sheaves, which explains the construction of such families in a geometric way. Since weight \(k\) classical modular forms (for \(k\) a positive integer) can be seen as global sections of \(\omega^k\) (where \(\omega\) is a certain line bundle constructed from the relative differentials of the universal elliptic curve over the open modular curve), the idea is to give a meaning to \(\omega^\kappa\), where now \(\kappa\) is a \(p\)-adic variable in the weight space, and show that \(\kappa\mapsto\omega^\kappa\) varies analytically.
The main purpose of this paper is to follow this geometric approach to the construction of the eigencurve outlined above to construct the Siegel eigenvariety, and \(p\)-adic families of Siegel modular forms. The Siegel eigenvariety constructed in this paper coincides with that constructed by E. Urban [Ann. Math. (2) 174, No. 3, 1685–1784 (2011; Zbl 1285.11081)] (with a different approach, as mentioned before) thanks to a result of G. Chenevier [Ann. Sci. Éc. Norm. Supér. (4) 44, No. 6, 963–1019 (2011; Zbl 1279.11056)]. Also, this geometric approach overcomes the restrictions allowed to above, arising when one tries to directly interpolated modular forms using overconvergent modular forms.
The main results of this paper give an elegant geometric construction of the Siegel eigenvariety (see in particular Theorems 1.1 and 1.2 of the Introduction for a complete technical account of the main results). The setting of Siegel modular forms presents new technical problems, due to the higher dimension of the eigenvariety. Besides the construction of families \(p\)-adic sheaves, the technical heart of the paper is to show that the specialization of a family of cuspforms at a \(p\)-adic weight \(\kappa\) is surjective onto the space of cuspidal overconvergent forms of weight \(\kappa\); the problem here is that strict neighbourhoods of the multiplicative ordinary locus in the toroidal compactification of the Siegel modular variety are not affinoids, so the authors need to descent their families of \(p\)-adic sheaves from the toroidal to the minimal compactification, where the image of strict neighborhoods are indeed affinoids.

MSC:

11F85 \(p\)-adic theory, local fields
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
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References:

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