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Problems in the theory of automorphic forms. (English) Zbl 0225.14022
Lectures in modern analysis and applications. III, Lect. Notes Math. 170, 18-61 (1970).
Let $$G$$ be a connected reductive algebraic group defined over a global field $$F$$. Let $$\mathbb A(F)$$ be the adele ring of $$F$$. $$G_{\mathbb A(F)}$$ is a locally compact topological group with $$G_F$$ as a discrete subgroup. The group $$G_{\mathbb A(F)}$$ acts on $$L^2(G_F\backslash G_{\mathbb A(F)})$$. Let $$\pi$$ be an irreducible representation of $$G_{\mathbb A(F)}$$ which occurs in $$L^2(G_F\backslash G_{\mathbb A(F)})$$. To a given $$G$$ the author introduces a complex analytic group $$\hat G_F$$ and to each complex analytic representation $$\sigma$$ of $$\hat G_F$$ and each $$\pi$$ he attaches an $$L$$-function $$L(s,\sigma,\pi)$$ defined by an Euler product of the local $$L$$-functions at “unramified” primes of $$F$$. Under some natural assumptions the author proves that the Euler product converges in a half-plane.
The author’s problems are mainly concerned with some fundamental properties of the $$L$$-functions:
– Are the $$L$$-functions meromorphic in the entire complex plane with only a finite number of poles and do they satisfy the functional equation of the usual form?
– Are there relations between the $$L$$-functions of different $$G$$?
– Is there a relation of the $$L$$-functions to the $$L$$-functions associated to non-singular algebraic varieties (especially for $$G= \mathrm{GL}(2)$$ and elliptic curves)?
The problems are posed in some reasonable precise manner. Some remarks are made about the cases where some of these problems are proved or may be proved $$(G= \mathrm{GL}(1). \mathrm{GL}(2))$$.
For the entire collection see [Zbl 0213.00101].
Reviewer: A. N. Andrianov

##### MSC:
 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F12 Automorphic forms, one variable 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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