Proceedings of the International Congress of Mathematicians
Helsinki, 1978
On Special Values of Zeta Functions of Totally Real
Algebraic Number Fields
Takuro Shintani
1. Hecke presented two different approaches to the Hilbert twelfth problem. One
is the study of higher dimensional complex multiplication which is now developed
to a magnificent theory by Shimura. Another is the study of Kronecker limit formula
for general algebraic number fields [4]. Namely he proposed that the study of analytic
expressions for values at s=l of the Hecke L-series for a given field k could lead
to a discovery of analytic functions suitable (natural) special values of which generate
abelian extensions of k (the author is indebted for the present view on Hecke's
works to Siegel [15] and Honda [7]). In the present talk we review our papers [11]-[14]
in the context of Hecke's second program.
Notation. For an algebraic number field k9 let hk9 dk and Oft be, respectively,
the class-number, the discriminant and the ring of integers of k. For each ideal
a of k9 N(a) is the absolute norm of a. The group of units (resp. totally positive
units) of k is denoted by E(k) (resp. L(/e)+). The 777th Bernoulli polynomial in
x is Bm(x).
2. Let k be an algebraic number field of degree n. For an integral ideal f of
k9 we denote by Hk(\) the group of narrow ideal classes modulo f of k. For
each c^Hk(\)9 Çk(s9c) is the ray class zeta function corresponding to c. For any
character % of Hk(\)9 Lk(s9x)=2%(c)£k(s>c) is the Hecke L-series associated
with %. For the arithmetic nature of Lk(l9x), Stark [20] conjectured that, if the
number of infinite primes of k where % splits is a(%)9 Lft(l,;t)7ia(*)""w would be
a homogeneous form of degree a(x) with algebraic coefficients in logarithms of
units of a certain abelian extension of k. In view of the conjecture, one may say that
an analytic expression for Lk(l9 %) of "desirable nature" should have a natural
592
Takuro Shintani
interpretation as a homogeneous form of degree a(x) in logarithms of special values
of certain analytic functions. In particular, if a(%)=Q9 "good formula" for Lk(l9x)
should be an elementary arithmetic one. Recall, when k is either rational or imaginary quadratic, classical formulas for Lfc(l, x) are all of "desirable nature" and
that special values of exponential or elliptic modular functions are involved in these
formulas. We will show that for a totally real field k9 an analytic formula of
"desirable nature" is available for Lk(\9x) if ct(x)=0 or 1.
3. Let A (resp. x) be an nXr (resp. lXn) matrix with positive entries. Denote
by Lm(z)=2am*zu ( a = l> •••>r) ^ e linear form in r variables corresponding to
the mt\\ row of A and set
(1)
Us9A9x) = 2
Z
È
Wl =
{Lm(z)+xm}-\
l
where the summation with respect to z is over all the Muples of nonnegative
integers. The Dirichlet series (1) is a natural generalization of the Riemann-Hurwitz
zeta function. It is convergent if Res>r/n and is extended to a meromorphic
function on C. Our starting point is that, if k is totally real, £>k(s9c) is a finite
linear combination of Dirichlet series of type (1). In the following we always assume
k to be totally real. Let x*-+(xa\ ...9xin))9 where *(1), ..., x{n) are n conjugates
of x9 be a natural embedding of k into Rn. By componentwise multiplication,
the group E(k)+ acts on R\. It is shown that a fundamental domain for Rn+
With respect to the action of E(k)+ is realized as a disjoint union of finite number
of open simplicial cones with generators in Ofc. In more detail, for r linearly independent vectors vl9 ...9vr of R"9 we call the set of all positive linear combinations
of vl9 ..., vr the r-dimensional open simplicial cone with generators vl9 ..., vr. It is
proved (cf. Proposition 4 of [11]) that there is a finite system of open simplicial
cones {Cj'9j£J} with generators in £)knR+ such that
(2)
Ä+ = U
U
uCj (disjoint union).
For each j£J9 we denote by r(j) the dimension of Cs and choose and fix a system
of generators vjl9 ..., vJHj)^Dk of Cj once and for all. For each xÇCj we set
x = 2 z(x)«Vj« (z(x)a£R+).
a=l
Note that if x£Cjnk9
ideals a^ ...,aAtf of k
narrow ideal classes of
(1</<A 0 ) such that c
jfê/, put
(3)
z(x)l9 ...9z(x)r(j) are all rational. Choose and fix integral
so that they form a complete set of representatives for
k. For each c£Hk(\)9 there uniquely exists an index i
and 0,-f are in the same narrow ideal class of k. For each
R(c9 Cj) = {xeCjnfaf)- 1 ; (x)û,fec, 0 < z(x\ < 1 (a = 1, ...,r(J))}.
Note that the set R(c9C}) is always finite. Let Aj be the nXr(j) matrix whose
(a, jS)-entry is given by i ^ . Then (2) and (3) imply the following formula for
On Special Values of Zeta Functions of Totally Real Algebraic Number Fields
W,e)
593
(cf. [11]):
(4)
N(*ifrtk(s, e) = 2
j£J
2
C(s,Aj,x).
x£R(c>Cj)
Note that a formula of type (4) was first introduced by £agier [24] in the case of real
quadratic fields.
Hence, evaluations of special values of £t(j, c) are reduced to those of £(s, Aj9 x).
Now, at non-positive integers, Ç(s9Aj9x) is evaluated in an elementary manner.
In more detail, let Lf(t)=2aait« ( a = l> —>w) be the linear form in n variables
corresponding to the rth column of A and let Bm(A9 x)^/(m\)n be the coefficient
of f/1^""1^!... tß_t tß+1... /j,)"1"1 in the Laurent expansion at the origin of the following
function in t and w:
[expf-ff 2Ux\
/7{l-exp(-ML*(0)H
Furthermore, set Bm(A9 x)=^2ßtslBm(A9 x)mjn. We may regard Bm(A9 x) as a generalization of Bernoulli polynomials. Applying a modification of the classical method
of contour integrals due to Riemann, Hurwitz and Barnes [2], we can prove (cf.
Proposition 1 of [11]) that
(5)
Ç(l-m9A9x)
= {-lY<m-»m-Bm(A9x)
(m = 1,2, ...).
Combining (4) and (5), we obtain an explicit formula for ttk(\—m9 c) (w=l, 2, ...)
(for real quadratic fields, such a formula was given in [17] by a different method)
which yields a simple proof of the Klingen-Siegel theorem [19] that £k(s9 c) is
rational valued at non-positive integers (cf. Theorem 1 of [11]). When f=Ofc,
an explicit formula for £fc(l —m9 c) of entirely different nature was established
in [18] (cf. [6]). Note that, when k is real quadratic, Cassou-Noguès [3] and Zagier
[25] also calculated Çk(l— m9 c) starting from the formula (4).
Now let x be a primitive character of the group Hk(f) which ramifies at all
infinite primes of k. Then the functional equation for Lk(s9 x) implies that
Lk(l, x) = Lk(09 X^Hx^/VdM),
where w(x) is the root number. Applying our formula for Çfc(0, c)9 we obtain
the following (cf. Theorem 3 of [11]):
THEOREM
(6)
1. The notation and assumptions being as above,
Lfc(l, x))/dkN(dl(n»w(x)) = 2 Z"1 (c)22
c
(ceflikd), j£J>
j
*i(C„ x)
X
xeR(cj9c))9
where
H
l
V«=l
'a=l
'a!J
where I ranges over all the r(j)-tuples of non-negative integers such that
!i+.-+Ir(» = r(j).
594
Takuro Shintani
In particular, assume x t 0 be of order 2. Then x corresponds, in class field
theory, to a totally imaginary quadratic extension K of k with relative discriminant
f and the left side of (6) is equal to 2n-lhK/hk[E(K)9 E(k)]. Thus, the formula (6)
gives an affirmative answer to the Hecke conjecture [5] that the relative class number
of K with respect to k would admit an elementary arithmetic expression. For
real quadratic fields k9 a formula equivalent to (6) was given by Hecke [5] and was
further studied by Meyer [8] and Siegel [16].
4. Next, we assume that % is a primitive character of LTfc(f) which splits at only
one of n infinite primes of k. Then the functional equation implies that
Lk(h x) = w(x)K(o9 z-WM-yiW®.
Thus, in the present case, the evaluation of Lk(l9 x) is reduced to that of £'(0, A9 x).
F o r a l X r matrix A and a variable x9 Barnes ([1], [2]) introduced multiple gamma
function rr(x9 A) with modulus A by setting
(7)
C'(0, A9 x) = log {rr(x9 A)lQr(A)l
1
where Q^A)" is the residue at x=0 of exp £'(0, A9 x). He established various
properties of Tr analogous to the ordinary gamma function. For a general nXr
matrix A with positive entries, £'(0, A9x) is also evaluated in terms of multiple
gamma functions.
In more detail, let Am be the /77th row of A and set, for each r-tuple / of nonnegative integers,
where (a, ß) ranges over all pairs of positive integers such that l ^ a , ß^n9
Furthermore, for *x=Az (z£Rr+)9 set
oc^ß.
where / ranges over all /'-tuples of nonnegative integers such that J-^+.^ + l^r.
Then it is proved (cf. Proposition 3 of [12] and Proposition 1 of [13]) that if *x=Az
(z£R\)9 Ç'(0, A9 ;c)=log T(x9 A). Thus, we are led to the following expression for
Lk(l>x) a s a linear combination of logarithms of special values of multiple gamma
functions (cf. Theorem 1 of [13]) :
THEOREM 2.
(8)
where
The notation and assumptions being as above,
w(x)-^dkN(VLh(l,
x)/(2n-i) =
m =n
2
JTHC)log TXc),
ij nx>A}).
Note that T(c) does depend upon the choice of representatives a l 5 ...,a, for
narrow ideal classes of k. For x of order 2, (8) yields a generalization of Dirichlet
class number formula for real quadratic fields.
On Special Values of Zeta Functions of Totally Real Algebraic Number Fields
595
Formula (8), together with conjectures of Stark, seems to suggest that multiple
gamma functions may play a significant role in construction of class fields over totally
real fields. In more detail let 5(f) be the subgroup of iffc(f) generated by those
principal ideals (x) with x=\ mod(f). Assume that there exists a character
X of 5(f) given by ^((x))=J7JL2sgn*(l)- I n P2], Stark introduced invariants
•finite) (m£Z> c^Hk(\)) by setting 8m(c)=exp{m2ses®*(s)C\0>cs)}- H e conjectured
{[21], [22], [23]) that cm(c) would be a unit of the abelian extension of k corresponding to the kernel of X and that the mapping c\-+sm(c) would be compatible with the
Artin canonical isomorphism. We can show
<9)
cm(c) = 77 W
w
".
Thus, the Stark invariants are described in terms of values at conjugates of k of
multiple gamma functions with modulus belonging to conjugates of k.
Stark's conjectures, which are strongly supported by his numerical experiments,
suggest also that Hecke's second program (mentioned in the introduction) would
ultimately work for totally real fields if a really good analytic formula for sm(c)
is available. At present one can only say that (9) may be a candidate for such a
formula.
5. For a real quadratic field /c, Theorem 2 is described in a simpler manner.
Take a fundamental totally positive unit e of k. Then we may put (cf. (2)),
\JCj = {x+ye; x>09y^0}.
Hence we may put (cf. (3))
UR(c,Cj)=:R(89c)
= {z = x+3;eÇ(a/.f)-1,(z)a/fGc, O^x^ì,
Set
=
O^y^
1,
x9y£Q}.
(1)
(I eod)
i
(l c«J-
Then (4) is simplified to
tf(û,D*Ckte, c) =
2
Us,A,z).
zeJR(e,c)
Assume that the group LTfc(f) has a character x which splits at only one of two
infinite primes of k. Take v£k such that v(1), v(2)>0 and v = - l (mod f).
Denote by the same letter v the element of #fc(f) represented by (v). Then
X(v)=~L Set
F(z9 (1, a)) = r2(z9 (1, £ ))/r 2 (l + £ - z , (1, e))
(cf. (7)). The function F is, up to a constant factor, the unique meromorphic
simultaneous solution of the difference equations F(z +1) = F(z)2 sin (TIZ/B),
F(z+e) = F(z)2 sin nz. It is shown (cf. Theorem 1 of [14])
C/(09c)-C(09cv) = logX1(c)9
596
Takuro Shintani
where
W =
77
F{z^9(l9^))F(z^9(l9B^)).
Let G be the kernel of x and put ^(c9G)^=JJgeGX^(cg). Then we have, if
X is primitive, the following simplified version of (8) :
L(h x)HxY1W;mi{2n) =
2
X-Kc)\ogX^(c9 G)9
c<Efffc(f)/<G,v>
where (G9 v) is the group generated by G and v.
Let KG be the class field over k corresponding to the subgroup G of Hk(\).
If KG is a quadratic extension of its maximally absolutely abelian subfield, Lk(s9 x)
coincides with an L-function of a suitable imaginary quadratic field. Applying
results of Ramachandra [9] on the arithmetic nature of Ljl9 x) for x imaginary
quadratic, we obtain the following result (cf. Theorem 2 of [14]) which is consistent
with the Stark conjecture.
THEOREM 3.
If KG is quadratic over its maximally absolutely abelian subfield,
then for a suitable positive integer m9 X^(c9 G)m is a unit of KG and the mapping
c\-+X^(c9 G)m is compatible with the Artin canonical isomorphism.
Theorem 3 implies that suitable multiplicative combinations of special values
of the function F(z9 (1, e)) at k do generate certain non-trivial abelian extensions
of k. However it remains quite mysterious why they do. It seems that most significant properties of the double (or multiple) gamma functions remain to be discovered.
We wonder if our results are related to Shimura's theory [10] on construction of
class fields over real quadratic fields.
Finally, we present a numerical example for Theorem 3. Set k=Q(Y2l)c:R.
The fundamental totally positive unit is given by e=(5 + ]/21)/2. Put f=(e— 1).
Then the group Hk(f) is isomorphic to the direct product of two copies of a cyclic
group of order 2. There uniquely exists the character x of Hktf) with a(x) = l
which splits at the infinite prime corresponding to the prescribed embedding of
k into R. Then KG=k(ye — 1). We may put a1=Dk9 a2=f. For this example,
Theorem 3 is valid for m= 1. We have
(10)
A't(l) = {F(l/3)F(l +8/3)F((2+2C)/3)}2
= (e-2+l/c~^I)/2,
where we put F(z)=F(z9 (1, c)) (cf. example 3 of § 3 of [12]). It would be quite
interesting if one could prove the equality (10) directly.
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On Special Values of Zela Functions of Totally Real Algebraic Number Fields
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UNIVERSITY OF TOKYO
7—3—1 HONGO, BUNKYO-KU, TOKYO 113, JAPAN