Automorphic Forms on GL(2) - H. Jacquet, R. Langlands
Formerly appeared as volume #114 in the Springer Lecture Notes in Mathematics, 1970, pp. 1-548
Two of the best known of Hecke’s achievements are his theory of L-functions with gr¨ossen-charakter, which are Dirichlet series which can be represented by Euler products, and his theory of the
Euler products, associated to automorphic forms on GL(2). Since a gr¨ossencharakter is an automorphic
form on GL(1) one is tempted to ask if the Euler products associated to automorphic forms on GL(2)
play a role in the theory of numbers similar to that played by the L-functions with gr¨ossencharakter.
In particular do they bear the same relation to the Artin L-functions associated to two-dimensional
representations of a Galois group as the Hecke L-functions bear to the Artin L-functions associated
to one-dimensional representations? Although we cannot answer the question definitively one of the
principal purposes of these notes is to provide some evidence that the answer is affirmative.
The evidence is presented in §12. It come from reexamining, along lines suggested by a recent
paper of Weil, the original work of Hecke. Anything novel in our reexamination comes from our point
of view which is the theory of group representations. Unfortunately the facts which we need from the
representation theory of GL(2) do not seem to be in the literature so we have to review, in Chapter I,
the representation theory of GL(2,F ) when F is a local field. §7 is an exceptional paragraph. It is not
used in the Hecke theory but in the chapter on automorphic forms and quaternion algebras.
Chapter I is long and tedious but there is nothing hard in it. Nonetheless it is necessary and
anyone who really wants to understand L-functions should take at least the results seriously for they
are very suggestive.
§9 and §10 are preparatory to the Hecke theory which is finally taken up in §11. We would like to
stress, since it may not be apparent, that our method is that of Hecke. In particular the principal tool is
the Mellin transform. The success of this method for GL(2) is related to the equality of the dimensions
of a Cartan subgroup and the unipotent radical of a Borel subgroup of PGL(2). The implication is that
our methods do not generalize. The results, with the exception of the converse theorem in the Hecke
theory, may.
The right way to establish the functional equation for the Dirichlet series associated to the
automorphic forms is probably that of Tate. In §13 we verify, essentially, that this method leads to the
same local factors as that of Hecke and in §14 we use the method of Tate to prove the functional equation
for the L-functions associated to automorphic forms on the multiplicative group of a quaternion
algebra. The results of §13 suggest a relation between the characters of representations of GL(2) and
the characters of representations of the multiplicative group of a quaternion algebra which is verified,
using the results of §13, in §15. This relation was well-known for archimedean fields but its significance
had not been stressed. Although our proof leaves something to be desired the result itself seems to us
to be one of the more striking facts brought out in these notes.
Both §15 and §16 are after thoughts; we did not discover the results in them until the rest of the
notes were almost complete. The arguments of §16 are only sketched and we ourselves have not verified
all the details. However the theorem of §16 is important and its proof is such a beautiful illustration
of the power and ultimate simplicity of the Selberg trace formula and the theory of harmonic analysis
on semi-simple groups that we could not resist adding it. Although we are very dissatisfied with the
methods of the first fifteen paragraphs we see no way to improve on those of §16. They are perhaps
the methods with which to attack the question left unsettled in §12.
We hope to publish a sequel to these notes which will include, among other things, a detailed
proof of the theorem of §16 as well as a discussion of its implications for number theory. The theorem
has, as these things go, a fairly long history. As far as we know the first forms of it were assertions about
the representability of automorphic forms by theta series associated to quaternary quadratic forms.
Formerly appeared as volume #114 in the Springer Lecture Notes in Mathematics, 1970, pp. 1-548
Two of the best known of Hecke’s achievements are his theory of L-functions with gr¨ossen-charakter, which are Dirichlet series which can be represented by Euler products, and his theory of the
Euler products, associated to automorphic forms on GL(2). Since a gr¨ossencharakter is an automorphic
form on GL(1) one is tempted to ask if the Euler products associated to automorphic forms on GL(2)
play a role in the theory of numbers similar to that played by the L-functions with gr¨ossencharakter.
In particular do they bear the same relation to the Artin L-functions associated to two-dimensional
representations of a Galois group as the Hecke L-functions bear to the Artin L-functions associated
to one-dimensional representations? Although we cannot answer the question definitively one of the
principal purposes of these notes is to provide some evidence that the answer is affirmative.
The evidence is presented in §12. It come from reexamining, along lines suggested by a recent
paper of Weil, the original work of Hecke. Anything novel in our reexamination comes from our point
of view which is the theory of group representations. Unfortunately the facts which we need from the
representation theory of GL(2) do not seem to be in the literature so we have to review, in Chapter I,
the representation theory of GL(2,F ) when F is a local field. §7 is an exceptional paragraph. It is not
used in the Hecke theory but in the chapter on automorphic forms and quaternion algebras.
Chapter I is long and tedious but there is nothing hard in it. Nonetheless it is necessary and
anyone who really wants to understand L-functions should take at least the results seriously for they
are very suggestive.
§9 and §10 are preparatory to the Hecke theory which is finally taken up in §11. We would like to
stress, since it may not be apparent, that our method is that of Hecke. In particular the principal tool is
the Mellin transform. The success of this method for GL(2) is related to the equality of the dimensions
of a Cartan subgroup and the unipotent radical of a Borel subgroup of PGL(2). The implication is that
our methods do not generalize. The results, with the exception of the converse theorem in the Hecke
theory, may.
The right way to establish the functional equation for the Dirichlet series associated to the
automorphic forms is probably that of Tate. In §13 we verify, essentially, that this method leads to the
same local factors as that of Hecke and in §14 we use the method of Tate to prove the functional equation
for the L-functions associated to automorphic forms on the multiplicative group of a quaternion
algebra. The results of §13 suggest a relation between the characters of representations of GL(2) and
the characters of representations of the multiplicative group of a quaternion algebra which is verified,
using the results of §13, in §15. This relation was well-known for archimedean fields but its significance
had not been stressed. Although our proof leaves something to be desired the result itself seems to us
to be one of the more striking facts brought out in these notes.
Both §15 and §16 are after thoughts; we did not discover the results in them until the rest of the
notes were almost complete. The arguments of §16 are only sketched and we ourselves have not verified
all the details. However the theorem of §16 is important and its proof is such a beautiful illustration
of the power and ultimate simplicity of the Selberg trace formula and the theory of harmonic analysis
on semi-simple groups that we could not resist adding it. Although we are very dissatisfied with the
methods of the first fifteen paragraphs we see no way to improve on those of §16. They are perhaps
the methods with which to attack the question left unsettled in §12.
We hope to publish a sequel to these notes which will include, among other things, a detailed
proof of the theorem of §16 as well as a discussion of its implications for number theory. The theorem
has, as these things go, a fairly long history. As far as we know the first forms of it were assertions about
the representability of automorphic forms by theta series associated to quaternary quadratic forms.
