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].

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) |