On the critical values of Hecke L

M ÉMOIRES DE LA S. M. F.
B ENEDICT H. G ROSS
D ON Z AGIER
On the critical values of Hecke L-series
Mémoires de la S. M. F. 2e série, tome 2 (1980), p. 49-54.
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SOCIETE MATHEMATIQUE DE FRANCE
2e serie, Mfimoire n0 2, 1980, p. 49-54
ON THE CRITICAL VALUES OF HECKE L-SERIES
by
Benedict H . GROSS
and
Don ZAGIER
Let
E
be the elliptic curve over
Q
with minimal model
y 2 + xy = x3 - x 2 - 2x - 1
.
The modular invariant, discriminant, and conductor of
E
are given by
3 - -33 . 53
N = (72)
Let
on
Q
E
.
dx
denote the fundamental real period of the Neron differential u) = ———
:
0 = J
(D = 1.93331170561681...
E(IR)
Over the field
<? = 7L\.—^""•~3
• Hence
0
Chowla and Selberg [l]
:
K = flK-^/^)
, E has complex multiplication by
can be determined explicitly, using an identity of
49
B . H . GROSS and D. ZAGJ^J?
^ ^ r(i/7)r(2/7)r(4/7)
^ . 2ni
Similarly, the L- series of
of
K . The conductor of
is prime to
7
E
X
is equal to the L-series of a Hecke character
is t"® ideal
(-/^) ; for
Q^ an ideal of
K
X
which
:
X(GO = a
where
0 = (a)
, a3 =
Krnod ^p?)
We have calculated the central critical values of the Hecke L-series
which are associated to odd powers of the character
X • Let
n ^ 1
be an
integer ; the Dirichlet series
Ux^s)^^2
^
s
IN 0
converges absolutely in the right half-plane
Re(s) > n •»• ^ . It extends -to a
holomorphic function on the entire complex plane : the modified function
A^2""1^) = (7/27c)s F(s) UX211''1^) satisfies Hecke's functional equation
ACX 2 "' 1 ^) = (-I)"'1 A(X2n-l,2n-s)
It follows that the value of
strip, vanishes when
When
n
n
Ux2""1,®)
at
s = n
:
.
, the center of the critical
is even.
is odd, define
a
n
by
_ ^2n-l ,
Q211"1
^__
L(X
n) « ————————r . .. , .
n l (n l)s
(2^/^) •
(We found this normalization by trial and error , it is consistent with the work
of Katz on the interpolation of real analytic
Eisenstein series [.2].)
The values of a for 1 ^ n ^ 33 are listed in Table 1 .
n
50
HECKE L-SERIES
Table 1
n
a
n
1
1/2
3
2
5
2
7
2(3) 2
9
2(7) 2
11
2(3 2 .5
13
2(3.7.2
15
2(3.7.3
17
2(3.5.'
19
2(3 3 .7
21
2(3 2 .5
23
2(3 3 .5
25
2(3 2 .7 S
27
2(3 2 .5
,
2(3 .5
29
2(3 5 .5
,
2(3 .5
31
33
Let p s Kmod 4) be a prime and let \
be the Hecke character
P
X (Q) = (wa) x(Q)
P
P
The Hecke L-series
L(x »s)
which becomes isomorphic to
.
is equal to the L-series of an elliptic curve
E
over
<(->/?)
51
. Let
0 = 0/^/p
P
and define
E /<0
a
n
by
B . H . GROSS and D. ZAGJ^R
^2n-l
"p
- / 2n-l .
L(x
n)
' • (W^)"-1 ^ •
-
Again, a p
n
vanishes when
(p)
n
n
is even. For
we found the values listed in Table 2.
52
n
odd and
p « 5> 13> 17> 29, 53
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B.H.
Note
:
GROSS and D.
ZAGJ£J?
In all the cases we computed,
twice a square, depending on whether
(?)
is
ap
-1 or
is either a square or
+1 . Can one prove this is
general ? Are there "higher Tate-Shafarevitch groups" associated to the abelian
varieties
(E ) "
whose orders can be conjecturally related to
P
these groups carry a natural alternating pairing ?
Bibliography
[l]
a
?
:
CHOWLA S . , and SELBERG A . , On Epstein*s Zeta Function. J . Crelle 227 ( 1 9 6 7 ) ,
96-110.
[2]
Do
"
KATZ N . , p-adic interpolation of real analytic Eisenstein series.
Annals Math. 104 ( 1 9 7 6 ) , 459-571.
B . Gross
Princeton University
Fine Hall
Princeton, N . J . 08540 ( U . S . A . )
D . Zagier
Mat. Inst. d . Univ. Bonn
Wegelerstrasse 10
53 Bonn ( B . R . D . )
54