Table Of Content1975 U.S. - JAPAN SEMINAR
APPLICATIONS OF AUTOMORPHIC FORMS TO NUMBER THEORY
Ann Arbor, Michigan
Abstracts and Summaries of Reports
by Participants and Observers
TABLE OF CONTENTS
Page
Tetsuya Asai
On certain Dirichlet series associated with Hilbert
'
modular forms and Rankin s me t h ~ d
John Coates
On the values of the p-adic zeta functions at the
odd positive integers
Stephen Gelbart
On the construction of modular forms of half integral weight
Larry Joel Goldstein
The arithmetic of Hecke differentials
Hir~akiH ijikata
On the theta series obtained from certain orders of
quaternion algebras
Ya sutaka Ihara
Some fundamental groups in arithmetic of algebraic
curves over finite fields
Masao Koike
On the congruences between Eisenstein series and cusp forms
Tomio Kubota
On a generalized Fourier transformation
R. P. Langlands
Shimura varieties and the Selberg trace formula
Erik A. Lippa
Hecke eigenforms of degree 2 : Fourier coefficients,
Dirichlet series, and Euler products
Yoshi nobu Nakai
On a Q-Weyl sum
Douglas Niebur
The Bessel function expansion of Fourier coefficients
of modular forms
Takayuki Oda
Construction of some transcendental 2-cycles on
certain Hilbert wdular surfaces
M ~ S ~i I IOIh trl
On reductions and zeta functions of varieties
obtaj ned from
TABLE OF CONTENTS
(continued)
Page
112 Arnold Pizer
Representing modular forms by generalized theta series
122 K. A. Ribet
Division points of J*(N)
131 Hiroshi Saito
Automorphic forms and algebraic extensions ~f
number fields
Kuang-yen Shih
On certain twisted modular equations
Hideo Shimizu
Some examples of new forms
Goro Shimura
On the special values of certain non-holomsrphic
derivatives of Hilbert modular forms
Takuro Shintani
On "liftings" of holomorphic automorphic forms
(a representation theoretic interpretation of
the recent work of H. ~ a i t o )
-
Non-Alphabetically-Arranged Supplement
Herve Jacquet
From G L ( ~ )t o G L ( ~ )
S. Lang and H. Trotter
Distribution of Frobenius automorphisms in
GL2-extensions of the rationals
A. P. Ogg
On the reduction modulo p of x ~ ( ~ M )
A. 0. L. Atkin
Modular forms of weight one, and supersingular
equations
ON CERTAIN DIRICHLET SERIES ASSOCIATED
WITH HILBERT MODULAR FORMS AND RANKIN' S METHOD
Tetsyya Asai
1, Let F be a real quadratic number field with the class number one and
the discriminant D, and @ be the ring of integers, Let
be a Dirichlet series ordinarily associated with a Hilbert modular cusp form
of weight k with respect to GL~(~?'),a nd so ~(02)'s are the Fourier
coefficients of the cusp form, where Ut runs over all non-zero integral
ideals of 6, It is well known that D(s) is holomorphically continued to
the whole complex s-plane and satisfies a functional equation,
Here we are concerned about a certain subseries of D(s), in which the
summation is restricted on the rational integers only, This series has
also the analytic continuation and a functional equation, Namely, if we put
and
-
C ,
where is the Riemann zeta function and s1 s-k+l then we get
Theorem 1. G*(s) can be holomorp-h ically continued t o the whole s-
plane except possible simple poles at s k, k-1, and'satisf ies the functional
-
equation G*(s) C*(2k-1-s) ,
This can be proved by so-called Rankin's method,
-1-
2. Let us assume that D(s) has the Euler product expansion
g w ,
where runs over all prime ideals of then we can obtain the Euler
product expansion of G(s), too. For this purpose, let us put
, $ ,
where the roots d depend on and so we can rewrite (3) as follows r
- -
k-l
Here pes are rational primes and we put V p*S ( and we also put v p
for later use ). Then we can see
Theorem 2. If D(s) has the Euler product expansion (4), then G(s)
has also its ~ulerp-r oduct expansion
3. Rankines method ( or convolution ) gives also the analytic continuation
of the Dirichlet series
,
(Re s ) k )
where SF is the Dedekind zeta function of F, s1 = s-k+l , and denotes
m.
the conjugate of In fact, the function
can be holomorphically continued t o the whole s-plane except possible simple
-
poles at s k, kbl, and satisfies H*(s) = H*(2k-1-S), Further, if ~ ( s )
has the Euler product expansion (4), H(s) has also its product expansion
There is a relation between G(s) and ~(s). Namely, under the
assumption that the discriminant D is odd, we can prove
Theorem 3. If ~ ( sh)a s the Euler product expansion, H(s) splits as
-
.
follows r ~ ( s ) G(s) G1(s) Here G (s), which is naturally defined by
1
the Euler product expansion, is also analytically continued and satisfies
a functional equation. In fact, both functions ~ ( s a)n d Gl(s) are holomorphic
-
but allowing the possibility that at most one of them has a simple pole at s k.
7
The holomorphy of G (s), which can be also expressed by the series
1
4. Lastly we give a remark on the case of the cusp form derived by Doi-
Naganma's map. There are at least two cases. Not only an elliptic
modular cusp form induces a Hilbert cusp form relative t o CL ( 8 )( Case 1,
2
(L)
say ), but also a cusp form of Neben type ( ( D ) ) induces a Hilbert
cusp form relative to C L ~w( ) ( Case /2 , say ). And we assume D(s) has
the Euler product expansion. Then we can see the followings. In Case 1,
-
G(s) is entire ( and Gl(s) has a simple pole at s k ), and in Case 2, G(s)
-
has a simple pole at s k ( and Gl(s) is entire ). Consequently, these
two cases are not overlapped each other.
The phenmena of theorems 1 and 2 were observed by Professor G. Shimura
almost a decade ago, and the author has completed some details only. The
author wishes to express his hearty thanks to Professor G. Shimura for his
suggestion and encouragement. The proofs of these and other details w i l l
appear elsewhere.
Department of Mathematics
Faculty of Science
Nagoya University
ON THE VALUES OF THE p-ADIC ZETA FUNCTIONS AT THE
ODD FOSITIVE INTEGERS
John Coates
Introduction. Let F be a totally real number field, and let G(F,S)
be the classical complex zeta function of F. The arithmetic meaning of the
values of G(F,S) at the odd positive integers remains one of the most
puzzling mysteries of number theory. Of course, it is classical that G(F,S)
has a simple pole at s = 1 with residue
,
where d is the degree of F over the rational field $ h is the class
number, Rw the regulator, and A the absolute value of the discriminant of
.
F over $ Recently, Kchtenbaum [8] asked whether K-theory might provide
.. . .
an analogous formula for <(F, s) at s = 3,5, More precisely, let 8
, ..
be the ring of integers of F and let Km B (m = 91,. ) be the K-groups
of 8 in the sense of Quillen [g] ; in particular, Ko8 is the ideal class
group and K @ the group of units of 8. For each integer m >- 1, let
1
w,(F) be the largest integer r such that the Galois group of F($) wer
F has exponent dividing m ; here is a primitive r-th root of unity.
cr
If A is a finite group, we write # (A) for the order of A . Then, for n
even and positive, Lichtenbaum asked (essentially) whether or not a formula
oi' the type
is valid, where ~,(n) is a generalized regulator formed from the torsion
! .
free subgroup of K2 n+lQ The precise definition of ~ ~ ( nis) du e to
Borel, who earlier had sho-i that K2,8 is finite, and that the torsion
I
%
free subgroup of n+l (9 has rank d (for n even and positive). More-
over, Borel has recently proven that the right and left hand sides of (1) do
in fact only differ by a multiple of a ratfonal number. But it seems un-
likely that his methods w i l l be able to determine this rational nwnber pre-
cisely.
LRt p be a prime number. In the present lecture, we propose a
p-adic analogue of Iiichtenbaumts conjecture, in which G(F, 1 +n) is replaced
by the n l u e of an appropriate p-adic L-function at l + n and ~,(n) is re-
placed by a higher p-adic regulator RP(n). That such an analogue might be
plausible is suggested by kopoldt 's theorem that, for abelian extensions F
,
of $ the p-adic zeta function of F has a simple pole at s = 1 with
residue
vhere R is Leopoldt 's p-adic regulator (see [6], $4 ), S is the set
P
.
of primes of F above p , and Np is the absolute norm of P One
suspects that the same result is true for a l l F, but this is unknown at
present. While we cannot prove this p-adic analogue of (1) in any single
case yet, we do give some evidence in favour of it by using Iwasawa's theory
of 2 -extensions of number fields. We should also point out that our
P
higher p-adic regulator is only defined up t o a p-adic unit, so that our
conjecture really only concerns the power of p occuring in the value of
the p-adic L-function at l + n .
,
l- 2-- adic regulators. Let 2 Qp be the p-adic integers and the p-adic
nuxibers, respectively. If a, #3 are non-zero p-adic numbers, a- f3 will
.
mean that a/#3 is a unit in ;I
P
As in § 1, F w i l l denote an arbitrary totally real number field. We
-
/fi ,
begin by giving an algebraic interpretation of R up to assuming
that R f 0. For each P E S , let U P be the units of the completion of
P
,
F at p and let U be the elements of U which are 1 mod p .
P,1 P
Then U is a profinite p-group, and hence a 22 -module. kt E be the
P,
1 P
,
units of the ring of integers (9 of F and let E c E be those units
.
which are 1 mod P for a l l p e S Let
