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Extended eigenvarieties for overconvergent cohomology. (English) Zbl 1444.11078
Summary: Recently, Andreatta, Iovita and Pilloni [F. Andreatta et al., Res. Math. Sci. 3, Paper No. 34, 36 p. (2016; Zbl 1417.11063); Ann. Sci. Éc. Norm. Supér. (4) 51, No. 3, 603–655 (2018; Zbl 1444.11075)] constructed spaces of overconvergent modular forms in characteristic \(p\), together with a natural extension of the Coleman-Mazur eigencurve over a compactified (adic) weight space. Similar ideas have also been used by Liu, Wan and Xiao [R. Liu et al., Duke Math. J. 166, No. 9, 1739–1787 (2017; Zbl 1423.11089)] to study the boundary of the eigencurve. This all goes back to an idea of R. F. Coleman [Invent. Math. 127, No. 3, 417–479 (1997; Zbl 0918.11026)].
In this article, we construct natural extensions of eigenvarieties for arbitrary reductive groups \(\mathbf G\) over a number field which are split at all places above \(p\). If \(\mathbf G\) is \(\operatorname{GL}_2/\mathbb Q\), then we obtain a new construction of the extended eigencurve of Andreatta-Iovita-Pilloni. If \(\mathbf G\) is an inner form of \(\operatorname{GL}_2\) associated to a definite quaternion algebra, our work gives a new perspective on some of the results of Liu-Wan-Xiao.
We build our extended eigenvarieties following Hansen’s construction using overconvergent cohomology. One key ingredient is a definition of locally analytic distribution modules which permits coefficients of characteristic \(p\) (and mixed characteristic). When \(\mathbf G\) is \(\operatorname{GL}_{n}\) over a totally real or CM number field, we also construct a family of Galois representations over the reduced extended eigenvariety.

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11F80 Galois representations
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