Table Of ContentLecture Notes ni
Mathematics
A collection of informal reports dna seminars
Edited by .A Dold, Heidelberg dna .B Eckmann, ZUrich
Series: Scuola Normate Superiore, Pisa
Adviser .E Vesentini
189
Andre Weil
Dirichlet Series
dna cihpromotuA Forms
Lezioni enaimreF
Springer-Verlag
Berlin. Heidelberg • New York 1791
AMS Subject Classifications (1970): 10D20, 12A85
ISBN 3-540-05382-4 Springer-Verlag Berlin • Heidelberg • New York
ISBN 0-387-05382-4 Springer-Verlag New York • Heidelberg • Berlin
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AVERT ISSEMENT
Invit~ par la Scuola Normale Superiore de Pise ~ y faire un cours
au printemps de 1969 dans el cadre des Lezioni Fermiane, y'j exposai
al th@orie dont j'avais d@j~ bri~vement indiqu~ les r@sultats ~ Bombay au
colloque de janvier 1968 {Bombay Colloquium on Algebraic Geometry,
T. .I F. R., Bombay 1968, pp. 409-426). Le but final en ~tait d'~tendre
tons les corps de nombres et de fonctions ("A-corps" o~ "corps
globaux") les r@sultats classiques de Hecke sur les transform@es de
Mellin des s~ries de Dirichlet, et plus particuli~rement le compl@ment
que y'j avais apportd dans une note de 1967 (Math. Ann. 168, pp. 149-156).
A Rise, li avait @t~ question de publier ce cours let qu'il avait dt~
,tiaf en italien. Mais j'en repris el sujet ~ Princeton en 1969-70, et une
rddaction provisoire en tuf alors pr~par~e par T. Miyake et H. Naganuma.
C'est en substance cette r@daction qu'ontrouvera ici, quelque peu
remanide te compl~t~e par l'adjonction des Chapitres VIII et XI. iI n'est
que juste ndanmoins qu'elle paraisse dans el cadre des Lezioni Fermiane,
puisque c'est ~ ce titre que j'eus l'occasion de traiter ce sujet avec
quelque ddtail pour al premiere lois. Je suis heureux d'adresser ici rues
remerciments chaleureux, avant ~ tout rues coll~gues de Pise, tout et
particuli~rement ~ E. Vesentini, pour leur invitation te la cordialit~ de
leur accueil, te @galement ~ T. Miyake te H. Naganuma pour leur
collaboration et el soin intelligent apportd ~ la r@daction de leurs notes.
Comme l'indiquais je d@j~ dans ma conf@rence de Bombay, les
recherches expos@es niec i font gu~re, ~ bien des ~gards, que doubler
une partie de celles de H. Jacquet te R. Langlands, qui forment el
contenu de leur monumental ouvrage Automorphic Forms on GL(2),
2 .bl 3 oz., Lecture-Notes No. 114, Springer 1970. Aussi jugera-t-on
peut-~tre que leur publication rendait celle-ci superflue, et cela d'autant
plus que leur point de rue (celui de al th@orie des representations) va
iv
sans doute plus au fond des choses que el mien, qui est essentiellement
41~mentaire. Mais c'est justement en raison de cette diffdrence de
points de vue, dans un domaine oh el dernier mot ne sera s~rement pas
prononc4 d'ici longtemps, que je n'ai pas cru ~ fair tout inutile de
mettre mes ddmonstrations ~ al disposition du public. Quant aux
"prioritds" li's( est quelqu'un qui s'y intdresse), li suffira de dire que
Jacquet te Langlands m' ont constamment tenu au courant de leur
travail, et qu'en plus 'd emprunts purset simples (par exemple tout ce
qui concerne l'"4quation fonctionnelle locale" pour GL(2, R) te GL(2, C)),
je n'ai pu manquer, consciemment ou non, d'en subir l'influence en
mainte occasion. Faute d'etre en 4tat de rien dire de plus precis, je
dois me contenter de leur exprimer ma reconnaissance, et de renvoyer
leur ouvrage (qu'on compl~tera utilement par R. Godement, Notes on
Jacquet-Langlands, I.A.S. 1970) el lecteur d4sireux de comparer leurs
r~sultats te les miens.
Enfin, c'est un agr~able devoir pour moi de remercier Miss
Margaret Murray ~ qui est due la dactylographie du pr4sent volume et
qui 's est acquitt4e de cette t~che parfois ~pineuse avec sa conscience
c outurni~ r .e
Princeton, el 26 novembre 1970.
TABLE OF CONTENTS
I, The classical case ............................... 1
.II Dirichlet series ................................. 9
.III Basic concepts .................................. 17
IV. The extension problem ........................... 23
V. The convergence lemmas ......................... 35
VI. Hecke operators ................................. 41
VII. Function-fields .................................. 47
viii.
Harmonicity at an infinite place ................... 63
IX. Harmonicity ( special cas e) ........................ 105
X. Number -fields ................................... 113
XI. Example s ....................................... 141
CHAPTER I
THE CLASSICAL CASE
.I Let G be a semisimple group, K sti maximal compact sub-
group, so that G/K si the associated Riemannian symmetric space. In
the classical theory of automorphic functions, one studies the functions
on G/K with a prescribed local behavior .e( .g holomorphic functions fi
G/K si Hermitian and si given sti "natural" complex structure), and a
prescribed behavior with respect to some suitable discrete subgroup F.
A typical case si given by the classical modular functions and modular
forms; there, G can be taken to be SL(2, ~); G/K si the Poincar~
half-plane; one may take for F the full modular group SL(2, ,)~ or a
congruence subgroup that of group. A rather full but somewhat longwinded
function-theoretic treatment that of case was given in 1890-92 by Fricke
and Klein; the arithmetical aspects, which are intimately tied up with the
theory fo complex multiplication, were considered by H. Weber in his
Algebra, vol. III a (for modern treatment, .fc a forthcoming book by
G. Shimura). The relation between modular forms and Dirichlet series
with functional equations was discovered by Hecke, whose epoch-making
work during the years 1930-1940, based on that discovery and that of the
"Hecke operators, " brought out completely new aspects a of theory which
many mathematicians would have regarded as a closed chapter long before.
Even in the earlier days of the theory, attempts were made to
extend ti to suitable types of automorphic functions of more than one
(complex) variable; an attempt by Hecke, in his early work, to extend
correspondingly the theory of complex multiplication was only partly
successful, for reasons which were not understood until much later.
Further inroads into the extension fo the classical theory to other groups
than SL(2, (cid:127)) had to depend upon a deeper understanding fo the
semisimple groups and of their classification; the most decisive steps were
taken by C. L. Siegel, in his famous work on the syrnplectic group, and by
H. Maass, who was the first to take up systematically the study of non-
holomorphic automorphic functions.
The role played by SL(2, E) in the theory fo modular functions had
already suggested to Hilbert the idea substituting of for E the ring of
integers ni a more general algebraic number-field; as long as holomorphic
functions were the only ones to be studied, this field had ot be taken totally
real, since otherwise the corresponding Riemannian symmetric space has
no complex structure. Even in that case, ti was soon noticed that serious
technical difficulties arose when the number fo ideal-classes si greater
than one; when the attempt was made to extend the work of Hecke to such
,sdleif ti became clear that the language and notations fo the classical theory
had become inadequate. a In parallel case, that fo Siegel's theory of
quadratic forms, similar difficulties had already been removed by a syste-
matic use of the adele concept, which makes ti possible to treat lla places
a of number-field simultaneously without unduly emphasizing some fo them
(the infinite or "archimedean" ones); this si also the best way of dealing at
the same time with number-fields and with function-fields, since the latter
have no infinite places. Naturally this does not mean that, in studying any
specific problem, one ought not to revert sometimes a to more traditional
treatment, where some places finite in number (including the infinite ones,
but not necessarily restricted to those) are singled out for special consider-
ation.
In these lectures, the adele point of view will be adopted systematically,
even though, at one stage (cf. Chapter the VIII) infinite places will have to
receive a separate treatment. The main result will be one which generalizes
quite (not completely) a theorem on modular forms, proved in my note of
1967 (loc. cir.) in continuation of Hecke's work. As the latter, naturally,
was stated in terms of the classical theory, the connection between them
cannot be perceived unless the means are given for shifting from one point
of view to the other; this will be done now.
.Z For technical reasons, we take as our starting point, not SL(g),
but the group G = GL(2), considered as an algebraic group over the
groundfield k = ~. With the notations which are now customary in the theory
of algebraic groups, k G si then GL(2, ~), and, fi v si any place of
k = ~, G denotes the group GL(2, k ;) in particular, we have
V V
G = GL(2, ~). For each natural prime ,p put ~I = GL(2, Z ;) this si
00 p p
the maximal compact subgroup of the group Gp = GL(2, ~p), consisting of
the matrices (: bd ) with ,a ,b ,c d in K (the ring of p-adic integers)
P
and ad - bc in KX (the group of p-adic units). The "adelized" group A G
P
si then the union (more precisely, the "inductive limit") of the groups
[-']" Gv×-]- T K ,
v(P P@p P
when one takes for P all the finite sets of places of Q containing .0o Then
k G si a discrete subgroup of G A-
From the adele point of view, the theory of automorphic functions,
or forms, on GL(2) over ~ si the theory of various types of functions on
GkNkGA, i.e. functions on A G which are left-invariant under k. G In a
loose sense, we may speak of the "space" of such functions (this can be
narrowed down in various ways, e.g. to LZ(Gk~GA ,) but we prefer not to
do ti here). In a loose sense again, this "space" may be "decomposed"
according to the characters, or rather to the quasicharacters, of the center.
Write ~ for the center of G as an algebraic group; this is isomorphic to
GL(1) and consists of the scalar multiples .z g 1 of the unit-matrix
iz = 0 . Then the center of A G consists of the elements
~- = .z ,zi where z si an idele of ~; ti may thus be identified, whenever
convenient, with the idele group ~A X of ~; the center ~ k of k G may
similarly be identified with X k -- ~X. Let ~ be a quasicharacter of ~A
J
.i( .e a morphism that of group into (cid:127)X); fi a function )q on Gk~G A
satisfies the condition
(1) ~p(g~) -- (p(g)a(~)
for lla A G ~ g and c ~ ~A' a must clearly be trivial .i( .e constant and
equal to )i on )k" Thus the "space" fo functions on Gk~G A may be de-
composed according to the quasicharacters of the idele-class group kA/kX;
to each such quasicharacter ,a we attach the functions )q on G A G
satisfying .)i( In particular, for = = ,i we get thfeu nctions on Gk\G A
which are invariant under ~A; as )A si central, we need not specify
here whether they are right-invariant or r'eft-invariant. As G/~ si the
same as the projective group G' = PGL(Z), such functions could also be
regarded as the functions on G~G k. In §§3-4, we will consider only such
functions.
.3 Call ° G the subgroup fo G given by det(g )> ,0 i.e. the
o0 0O o0
connected component 2 of in 1 G ", call ~ the open subgroup ° G <) ~I<i
00 0o p
fo G A. Then we have A G = Gk~. In ,tcaf g > det(g) si a morphism fo
A G onto QA' X and QA X si the direct product of ~× and fo the subgroup
+X~ X-~Z pX of ~A'× For any c g GA, write det g r.u, = with E r QX,
u > ,0 u ¢ EX for lla ;p put
0o p p
'g = "g" 0 i ;
then det 'g = .i In view fo the well-known "approximation theorem" for
SL(2, Q), we can find Y1 ~ SL(2, ~) such that, for lla ,p the p-component
of ¥i ,gI si in K . p Put y = 0 'i/~" then y si in k G and ¥ g in ,~
which proves our assertion.
Consequently, a function ~q on GkXG A si uniquely determined by
sti values on ~. Conversely, a function on ~ can be extended a to function
on GA, left-invariant under Gk, fi and only fi ti si left-invariant under the
group k A G ~, which is no other than the "classical" modular group
SL(2, Z). In the same manner, we see that #~ is invariant under the center
of A G fi and only fi the function ti induces on ~ is invariant under the
center of ~.
Now we restrict our function-space again, by prescribing that
shall be right-invariant under all the groups K , and prescribing also sti
P
behavior under the maximal compact subgroup of GO; this consists of the
oc
rotations
cos @ sin @ )
r(@) = k_si n @ cos @
We do this by assuming
ui @
)2( ~#(g.r(8)) ~#(g)e =
for lla A G g ~ and all .@ As we have agreed, in §2, to take a = ,i
must be an even integer, since r(~) = -l .2
As one sees at once , every element g of ° G can be uniquely
co co
written in the form
)3( ocg = .z (0 ~). )e(r
with z > ,0 x > ;0 in view of what we have found, this shows that ,#¢
subject to the above conditions, is uniquely determined by the function
f(x, y) ti induces on the subgroup ° B of ° G consisting of the matrices
o( 0 00 oo
x>
O
4. Let now a function f(x, y) be given on B ; we wish to know
co
whether ti can be extended to a function #~ on GA, of the kind described
in §§2-3. Take g c ~, and write g in the form (3). Then we must have
co
)g(pc = f(x, y)e @ iv
