# Automorphic Forms on GL(2)

@inproceedings{Langlands1970AutomorphicFO, title={Automorphic Forms on GL(2)}, author={Robert P. Langlands}, year={1970} }

In [3] Jacquet and I investigated the standard theory of automorphic forms from the point of view of group representations. I would like on this occasion not only to indicate the results we obtained but also to justify our point of view. For us it is imperative not to consider functions on the upper half plane but rather to consider functions on GL(2,Q)\GL(2,A(Q)) where A(Q) is the adéle ring of Q. We also replace Q by an arbitrary number field or function field (in one variable over a finite…

## 1,277 Citations

### On Automorphic Forms on GL 2 and Hecke Operators

- Mathematics
- 1971

In [4], Jacquet and Langlands showed that if an irreducible unitary representation of the general linear group of degree 2 over the adele ring of an A-field' is an irreducible constituent of the…

### Automorphie Forms on Covering Groups of GL(2)

- Mathematics
- 1980

The theory of modular forms of half-integral weight has seen much progress since 1973 when Shimura [14] associated to each cusp form of weight 89 (odd k>3) and character Z a modular form of weight k…

### $L$ functions for the group $G_2$

- Mathematics
- 1990

The method of L functions is one of the major methods for analyzing automorphic forms. For example, the Hecke Converse Theorem gives an equivalence via the Mellin transform between holomorphic…

### On certain zeta functions attached to two Hilbert modular forms II. The case of automorphic forms on a quaternion algebra

- Mathematics
- 1981

As to the meaning of the symbols, the reader is referred to the introduction of Part I. We recall here only that F is a totally real algebraic number field of degree n, and w denotes the Fourier…

### On the decomposition of a representation of $GL(3)$ restricted to $GL(2)$ over a $P$-adic field

- Mathematics
- 1993

points ofthe corresponding algebraic group over a fixed nonarchimedean local field k. In this paper we give a complete classification of those irreducible admissible representations V of GL(3), and W…

### ON THE ESTIMATION OF EIGENVALUES OF HECKE OPERATORS

- Mathematics
- 2008

1. Introduction. Let A denote the adele ring of the rational numbers Q and suppose that we have a cuspidal automorphic representation re of GL 2 (A). (For the terminology and the details, we refer…

### BASE CHANGE FOR GL(2)

- Mathematics
- 1980

R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, ?) attached to a cuspdidal automorphic…

### A global approach to the Rankin-Selberg convolution for (3,)

- Mathematics
- 1987

We discuss the Rankin-Selberg convolution on GL(3, Z) in the 'classical' language of symmetric spaces and automorphic forms. Introduction. Study of the Rankin-Selberg convolution of two automorphic…

### Non-vanishing of L-functions of 2 × 2 unitary groups

- Mathematics
- 1993

Let π be a cuspidal automorphic representation of GL(2, E), E a totally real field, and let Jf be a CM quadratic extension of E', let π^ denote the base change of π to JT. We show, using the…

### On some arithmetic properties of automorphic forms of GL(m) over a division algebra

- Mathematics
- 2011

In this paper we investigate arithmetic properties of automorphic forms on the group G' = GL_m/D, for a central division-algebra D over an arbitrary number field F. The results of this article are…

## References

SHOWING 1-10 OF 25 REFERENCES

### ON DIRICHLET SERIES AND ABELIAN VARIETIES ATTACHED TO AUTOMORPHIC FORMS

- Mathematics
- 1962

Since Hecke [13] had given a general theory of constructing Dirichlet series with Euler-product and functional equation out of elliptic modular forms of any level, several authors considered its…

### ON DISCONTINUOUS GROUPS OPERATING ON THE PRODUCT OF THE UPPER HALF PLANES

- Mathematics
- 1963

Let H" be the direct product of the n upper half planes, and let G be the connected component of the identity of the group of all analytic automorphisms of Hn. G is the direct product of n subgroups…

### Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series

- Mathematics
- 1956

In The following lectures we shall give a brief sketch of some representative parts of certain investigations that have been undertaken during the last five years. The center of these investigations…

### Sur la Théorie du Corps de Classes

- Mathematics
- 1951

© Association des collaborateurs de Nicolas Bourbaki, 1951-1954, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. ens.fr/) implique l’accord avec les conditions…

### On Spherical Functions over p-adic Fields

- Mathematics
- 1962

A o-adic analogue of the spherical function was first considered by Mautner in the case of PLy. and then by Tamagawa for GLn over any o-adic division algebra. The purpose of the present note is to…

### On Zeta Functions of Quaternion Algebras

- Mathematics
- 1965

The present paper is concerned with some equalities between zeta functions of quaternion algebras introduced in Godement [6], Shimura [11], Tamagawa [13]. Let A be a quaternion algebra over a totally…