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On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms. (English) Zbl 1090.22008

Let \(\mathbb G\) be a connected reductive linear algebraic group defined over a number field \(F\). Let \(\mathfrak{p}\) be a finite prime of \(F\). The author considers the group \(\mathbb G(\mathbb A_f)\), where \(\mathbb A_f\) is the ring of finite adeles of \(F\), and constructs a family of locally analytic representations on locally convex topological vector spaces over a finite extension of \(\mathbb Q_p\). These representations are used to obtain \(\mathfrak{p}\)-adic analytic families of systems of Hecke eigenvalues, which \(\mathfrak{p}\)-adically interpolate the systems of Hecke eigenvalues attached to automorphic representations of cohomological type [see R. Coleman and B. Mazur, Lond. Math. Soc. Lect. Note Ser. 254, 1–113 (1998; Zbl 0932.11030)]. The case \(\mathbb G=GL_2\), \(F=\mathbb Q\) is considered in detail. A survey of the related literature is given.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11R56 Adèle rings and groups

Citations:

Zbl 0932.11030
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