A construction of the full eigenvariety of a reductive group
Eigenvariety is a geometrical object parameterizing \(p\)-adic analytic families of overconvergent automorphic forms on reductive groups. It is a generalization of the eigencurve constructed by \textit{R. Coleman} and \textit{B. Mazur} [Lond. Math. Soc. Lect. Note Ser. 254, 1--113 (1998; Zbl 0932.11030)] in the case of \(p\)-adic families of cuspidal modular eigenforms. In a recent paper, \textit{E. Urban} [Ann. Math. (2) 174, No. 3, 1685--1784 (2011; Zbl 1285.11081)] constructed eigenvarieties for reductive groups defined over \(\mathbb{Q}\) such that its Archimedean part has discrete series. His eigenvariety parameterizes \(p\)-adic overconvergent cohomological automorphic representations with non-trivial Euler-Poincaré characteristic.NEWLINENEWLINEThe paper under review describes a construction of the full eigenvariety for any reductive group defined over \(\mathbb{Q}\), that is, an eigenvariety parameterizing all \(p\)-adic overconvergent cohomological automorphic eigensystems. The method is to consider the so-called overconvergent cohomology, that is, the cohomology of arithmetic groups with coefficients in certain universal distribution spaces.NEWLINENEWLINEIn the unpublished notes of \textit{A. Ash} and \textit{G. Stevens} [``\(p\)-adic deformations of arithmetic cohomology'', preprint, 2008, see also A. Ash, Acta Arith. 149, No. 4, 305--319 (2011; Zbl 1242.11037)], they construct local eigenvarieties by making the slope decomposition of the overconvergent cohomology. Although these local pieces contain slope information, they cannot be patched together.NEWLINENEWLINEHence, the author makes a more general polynomial decomposition of a certain perfect complex computing the overconvergent cohomology. In that way the local pieces may be easily patched together to form the full eigenvariety, but at the price of losing the information on classical points. For instance, the slope information is lost, the components may not contain a Zariski dense subset of classical points, and the eigenvariety is not equidimensional.
scientific article; zbMATH DE number 6030252