## 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

Zbl 0932.11030
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### References:

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