•
Lecture Notes In Mathematics
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Arithmetic on Elliptic Curves with Complex Multiplication
II,
With an Appendix by B. Mazur
~
Springer-Verlag 5 9 <j c; Berlin Heidelberg New York 1980
=
Table of Contents
o.
Introduction
1.
Acknowledgements
1 2
2.
Notation
3
Chapter 1:
3. 4. i
7.
8. Chapter 2:
9. 10. 11.
12. 14.
4:
17.
18. 19. 20.
Chapter 5:
21. 22. 23.
24. Appendix
, ,
I f
t
20
29 32
Local arithmetic
34 38 42
Global arithmetic
15. Restriction of Scalars 16.
12
14 17
23
A classification over F A rational p-isogeny Local invariants and global torsion
13.
4 8
A classification
Curves over H . Descended curves Ill-curves.
Chapter 3:
Chapter
The theory of complex multiplication Elliptic curves Elliptic curves over II: and E The analytic theory of complex multiplication Elliptic curves over p-adic fields ~adic Galois representations. The arithmetic theory of complex multiplication
5. 6.
,
and Conventions
The Ill-rank The first descent . A factorization of the L-series The sign in the functional equation tQ..... curves and modular forms . The Ill-curve
45 49 53
57 60 64
A(p )
67
Periods The rank of A(p) • Global models Computational examples
72
80
82
(by B. Mazur)
25.
The cohomology of the Fermat group scheme
87
26. 27.
Bibliography Index.
92 94
•
O.
Introduction.
K be an imaginary quadratic field, with ring of integers
Let number
h
and let
A be an elliptic curve over
Let
j (A)
of degree
h
be the modular invariant of
over
Sk~d
Then
j (A)
This fundamental result has its practical drawbacks. cannot be defined over
~
0,
is an algebraic integer
its conjugates generate the Hilbert class-field
~
class-
with complex multiplication by
~
A.
0
when the class-number of
H of
K.
For example, the curve
K is greater than one.
A
We can
often circumvent this problem by passing to the category of elliptic curves up to
isogeny.
of curves defined over their field of moduli all of their Galois conjugates
they
~
K is odd, one has a large supply
Specifically, when the discriminant of
defined over
Ql.
~
.!!.
F
=
Ill(j (A))
which are isogenous to
Arithmetically these curves behave as if
I call them (l-curves, and these notes are devoted to
their study 1
In Chapter 1 we recall some of the general theory of elliptic curves with complex. multiplication.
The treatment will be brief:
this sUbject have already appeared in print. curves' A over scend to
F
In Chapter 2 we classify elliptic
H with complex multiplication by
= Ill(j (A»
many excellent references on
0
We show which curves de-
and which descended curves are actually Ill-curves.
In Chapter 3, we study the arithmetic of descended curves at all completions of the field nant
-p.
F.
For simplicity, we restrict to the case where. K
In Chapter
4 we investigate the global arithmetic of Ill-curves.
5 is devoted to a detailed study of the ill-curve integers
!
Chapter
A(p) • with multiplication by the
0 of III( ,cp) and good reduction at all places of F
We end with a discussion of some questions which remain open.
I
has prime discrimi-
not dividing
p.
1.
Acknowledgements.
It is a pleasure to acknowledge the mathematical assistance I received from Joe Buhler, Pierre Deligne, Ken Kramer, Barry Mazur, Gilles Robert, David Rohrlich,
Jean-Pierre Serre, and Don Zagier. cellent job of typing.
,
t
I also wish to thank Lauri M. Hein for the ex-
Much more than thanks are due my family and friends --
Debby Gans, Ian Morrison, and Jane Reynolds -- who kept me distracted during the write-up.
Finally, I want to thank my teacher, John Tate, for all the inspiration
and support he gave me in the course of this work. M
r
fiI :J
no· f
Princeton, New Jersey
July, 1979
ga
2.
Notation and Conventions.
pm
Groups
G will always act on the left.
m~
action will be written either as
Lch,
sub-module of G-invariants.
e:x-
action of
a
If
and
M
On a homomorphism
u(m)
f:M
If
M is a G-module and
or
m t----+
um
E
G , this
G M denote the
We let
are G-modules, so is
N
a
Hom(M,N) :
the
is given by
--+ N
je ion
Rings will also act on the left. M r
=
{m EM: rm
=
If
B
A and
O}
fined over the field
If
M is an R-module and
we write
(fA
Ho~(A,B)
F , we let
and
gate homomorphism from
t
If
F
R
we let
are elliptic curves (or, more generally, abelian varieties) de-
F.
If
be the group of algebraic homomorphisms S
is any F-algebra, we let
note the abelian group of all S-rational points of F
g
denote the sub-module of "r-torsion."
$:A --+ B which are defined over
of
r
is a field,
If
a
for the conjugate varieties, and
(fB
crA to
F
A
A(S)
de-
is any automorphism a$
for the conju-
cr B .
denotes an algebraic closure of it.
We shall
alw~s
use
i
the isomorphism of local class field theory which takes a uniformizing parameter to
I
an arithmetic Frobenius in the Galois group.
•
I t
Chapter 1
3.
Elliptic curves 3.1.
The theory of complex multiplication
Deligne [6], Tate [29]).
(References:
An elliptic curve
singular curve of genus
A over the field
lover
F
F, furnished with a F-rational point
the theorem of Riemann-Roch, there exist functions on
is a complete, irreducible,
A which are regular outside of
0A'
x
and
y
of degree
0A 2
000-
By and
3
These functions, when suitably normalized,
Then we
satisfy an equation
'h
where the coefficients
Weierstrass model for
lie in
A.
Then
x
We call such an equation a generalized
F.
and y
generate the function field
F(A) ,
fine~
b
inva:""":B
and the above model is unique up to a change of coordinates of the form: 2
x' = u x + r in
where
u
is in F*
and
r,s,t
are in F.
Associated to the model (3.1.1) we
have the non-Zero differential of the first kind
This gives a basis for the F-vector space
HO(A,Ol).
Under a change of coordinates
£&)1
menta
t
variant
(3.1.2) We find
3.2.
Henc
=
U
-1
•
to •
Given a generalized Weierstrass model for
A over
F. define the ale-
3. beeD
5
(3.2.1)
= a 12
4
= a1a 3
6
= a 32
8
= b 2a 6
b b b
non-
+ 4a
2
b
c4
= b 22
c
6
= _b 23
= -b 2b 8
8b 3 4
2
+ 2a4
+ 4a
6
6
2
_ 24b
4
+ 36b b - 216b 2 4 6
27b
2 b b b 6 + 9 2 4 6
2 2 - a a a4 + a a - a4 2 3 1 3
3
Then we have the relation
The condition that
tJ." 0
is equivalent to the assertion that the curve de-
fined by (3.1.1) is non-singular.
This being the case, we may define the "modular
invariant It :
Under a change of coordinates (3.1.2) we find (3.2.4)
;es
Hence the quantities variant
3.3
j
= j (A)
If
c4' c6
and
tJ.
c4 •
= u 4c 4
c6 •
= u6c 6
tJ.'
= u 12tJ.
j
=j
I
depend only on the pair
depends only on the curve
A and
become isomorphic over
B are two curves ov"r
F.
. A.
F
If
with
a
is any automorphism of
j (A)
More generally, assume that
(A,.. ) , and the in-
F
= j (B)
, then
A and
F:
B
is a perfect field and
6
G = Gal(F!F)
let
Then there is a bijection between the pointed sets
B/F} +---+ ~(G,AutF-(A»)
{Isomorphism classes of with j(B) = j(A)
which takes If
A to the trivial class.
j(B) = j(A)
This bijection is constructed as follows.
we may choose an isomorphism
$:A ~ B
over
F.
The
H
assignment ~:G ---+ Autp(A)
a>---> $-1
is a continuous l-cocycle on phism class of
B =
B.
0
a$
G whose cohomology class depends only on the F-isomor-
Conversely, given such a cocycle
tIJ
one can construct a "twist"
A~ over F with j(B) = j(A) B(F) = ip
The isomorphism class of
3.4.
B
over
Since the curve
A
a(P) =
A(F)
E
F
~(a)
0
p} •
depends only on the cohomology class of
is isomorphic to its Jacobian over
the structure of an abelian variety.
The distinguished point
the identity in the algebraic group.
Any non-zero homomorphism
called an isogeny.
The (separable) degree of
corresponding field extension: The set law on
B.
Ho~(A.B)
The group
[F(A): $
0
En~(A)
0A
corresponds to
$:A --+ B
is
is the (separable) degree of the
F(B)]
forms a ring with multiplication given by the com-
For any
m
E ~
plication by m" in the lalgebraic group of
PA
F, it inherits
inherits the structure of an abelian group from the addition
position of homomorphisms'.
separable i f and only if
$
~.
m
A
is prime to· char (F)
IIllIY have separable degree either
ordinary, in the second case
we let
A
p
or
0
m be the endomorphism "multiA 2 This isogeny has degree m and is If
char (F)
=p
In the first case
, the isogeny A
is said to be
is said to be supersingular.
rt
' ~..•.
lC
7
If
$:A --+ B
defined over
F
is any isogen>' of degree
m, there is 'a dual isogeny
v-
$:B --+ A
with v~ 0
$ =
$
v~ =~
0
ffi
A
Hence the relation of isogeny is an equivale~e relation on the set of curves over
Jmor-
dst"
s
e
,ion 1-
is
be
F.
8
4.
Elliptic curves over 4.1.
Let
A
n:
and m
(Reference:
be an elliptic curve over
~.
gives a closed I-form on the Riemann surface
Weil [33J). Any differential
A(n:).
w
g
o 1 H (A,n )
(4.1.:'
w # 0 , its set of inte-
If
Given
gral periods W = {f w
(4.1.1)
Y
q.
(4.1. 6)
HI (A(n:),Z)}
£
i
j
y
forms a lattice in
l:c, and the map
A(n:) r--J n:/w
(4.1.2)
(mod W) (B,
If
is an analytic isomorphism.
(4. Ie ,')
Conversely, given any lattice
W in
n: , let The
(4.1.3)
g2(W)
=
60
L
-4 w
= 140 L
~g
are -:'10
wow
f
W#o g3(W)
I
l"
-6
homo_.1E
w
woW
W#o
4.
Jacc t'
These series are both convergent; define a complex curve
~
by the equation, q
= e 21
I
(4.1.4)
(4.: 1 This curve is elliptic, and Weierstrass I s parametrization gives an analytic isomorphism
E/W ~ ~(n:) z >--+- Ci'w(z),RT' (z» = (x,y) .
The holomorphic diff'erential
"' = dx y
pulls back to the differential
and W is its lattice of' integral periods.
dz
on
are
IQ
(4.
2
Thi
c
n:,
9
This establishes a bijection between {pairs(A,w)!~}
(4.1.5)
Given a lattice
W, the invariants of
(A,w)
c
6
= 216 g3(W)
11 = g2 (W)
- 27 g3 (W)
corresponding to the lattice
HO~(A,B) = {a €~.
(4.1. 7)
2
3
~
is another pair over
3
= c4!11
j
(B,v)
are given by
c4 = 12 g2(W)
(4.1.6)
If
W ~ ~}
+---+ {lattices
The degree of the isogeny corresponding to
a
V , one has
aW ~V} .
[V: aWl.
is the index
Two curves
are isomorphic iff their lattices (with respect to any choice of differentials) are
homothetic.
4.2.
It is often convenient to convert from the language of lattices to
Jacobi's q-parametrization. q=e 2wiT (4.2.1)
Iq I
Then
0 , set
10
(4.2.3)
c4
= E4(q)
In boi
scalar.
c6 = -E 6 (q) t.>.
= '1. •
IT (1_qn)24
v
n>l If
(4.2.4)
A (~) , then '1.
w on
W is the lattice of integral periods of
(4.3.4)
W = 27Ti (Z fl ZT) .
Hence
The exponential map gives an analytic isomorphism
(4.2.5) z 1--+ e
4.3.
z
Tate has observed that Jacobi's parametrization gives a simple analytic
description of elliptic curves over lR .
Proposition with
'1.
4.3.1.
real and
!1:22!..
0
0 • A
q
if
'1.
is real and
is then Re(w) = 0
21Ii
(2w-l) (Z tl b )
I
11
In both cases, we see that
W
is homvthetic to
q
W via multiplication by a real
scalar. But for any two pairs
(A,w)
and
(B,,,)
oyer 1R
with period lattices
Wand
v (4.3.4) Hence
1 2
{" £
A
is JR-isomorphic to
A
q
1R: "W S;; v} .
The uniqueness of
g
may be checked similarly.
12
5.
The analytic theory of complex multiplication
(References,
5.2.
Lang [12], Shimura
[22]) .
actly
criminant 5.1.
Let
A be an elliptic curve over End~(A)
cation if the ring field
K.
[.
We say
is isomorphic to an order
In this case, the lattice
A
R
has complex multipli-
in an imaginary quadratic
W of periods of any
non- zero
differential
w is a projective R-module of rank 1. Assume that Then
End~(A)
0,
is isomorphic to
the full ring of integers of
K.
where
W has the form
modulus:
(5.1.1)
n
where
£ ~
and
~
is a fractional ideal of
determined by the image of morphism of j (A)
~, the curve
has at most In fact,
j(A)
h
~
K.
The isomorphism class of
in the ideal class-group of
K.
If
a
aA also has complex multiplication by O.
A
is any autoConsequently
Aut(~) , where
h
is the class number of
is an algebraic integer of degree
h
over
conjugates under
is the Hilbert class-field of
K.
is
~, and
H
The sec:
K.
= K(j(A»
If we identify the groups
Bc...l cl(K) -
Gal(H/K) points
via the Artin isomorphism, then the Galois group permutes the conjugates of
,]
j (A)
as follows:
Here we write
J ( !!. ) for the complex modular invariant of the curve 0:/$.
One
also has the formula for complex conjugation:
(5.1.4)
which gives .the full action of
Gal(RfIll) •
I
13
;ura
I J
5.2. actly
2
criminant
By (5.1.4), the modular invariant
.j(O)
is real.
In fact, there are ex-
real isomorphism classes with this modulus (by (3.3.1».
-D
of
When the dis-
K is odd, these curves correspond to the lattices
li-
ftic I
;ial
(5.2.1)
where
-I
V
=
is the inverse different of
K.
The first curve has real
modulus:
'4
I
= -e
-w/ID
is
jtoThe second has modulus
Both
curves have one component in their real locus.
points where
dy/dx = 0
On the latter, the four
are real, so the graph looks pinched.
A
'4
14
6. /Elliptic curves over p-adic fields
6.1.
F be a
Let
formizing parameter
F* so that
of
Let
~inite
n , and residue field
=1
v(n)
with ring of integers
k = R/nR.
A
A
over
k.
An equation (3.1.1) for
F.
a.
all lie in
1
in
R
(if
A
A
over
F is
A
A
of
F.
multiplication: for such curves
I
l
(mod n)
gives the equation
m
F
F.
•
A ns v(c4 )
If
v(t.) > 0
=0
the reduced curve
which is isomorphic either to
=0 • then
A
; in
In this case the non-singuOl
AC(F
A(
I.
fo
= c~/t.
j
A1 "
st,...u
a
v(j) < 0).
so
its modular invariant
v(j) ~ 0
will neces-
acquires good reduction over
This will alNs be the case when j (A)
v(t.)
is an algebraic integer.
A
When
has compleX v(J) ~ 0
we
have the inequality
(6.1.1)
(6. .
is minimal subject
is elliptic if and only if
(if
Ol
Conversely, if E
A
has bad reduction.
has good reduction over
a finite extension
0
The isomorphism class of this curve is independent of
form an algebraic group
sarily be integral.
on
(6.2.
A has good reduction over
v(c4) > 0) or to a form of If
v(~)
and
A
R.
The curve
has a singularity and we say
lar points of
v I
r, s, t
the minimal model chosen. this case we say
Normalize the valuation
!'
Reducing the coefficients of a minimal model for of a cubic curve
(6.
uni-
R ,
Such a model is unique up to a change of coordinates (3.1.2)
* Rand
in
Tate (28).
.
A be an elliptic curve over
to that condition.
u
~p'
extension of
called minimal if the coefficients
with
(Reference:
g: .•
the
it
o ~ v(t.)
< 12
+ 12v(2) +
~(~)
for the exponent of the disct'1minant of any minilbalmocie1. >. One Cao also associate to
A/F
the exponent
v(tl)
of the conductor.
This nClrl~ Fur
negative integer is ao isogeny invariant which measures the amount af wild ramification in the division fields of
over
F.
A.
! t i s zero if and only i f
A has good reduction
I
I ,
i
15
6.2.
We can define a filtration:
(6.2.1)
A(F) 2Ao(F)
2
A (F) '2.· .. 2 nAn(F) = (a) l n=O
v on the p-adic Lie group is
A{F)
as follows.
Let
-
(6.2.2)
PEA (k)} ns
ect and for
~
n
1
let
vex)
(6.2.3)
< -2n
v(y)
and
~ -3n}
tion
J
G of
I I
1 ~
1-
!
r r
where
x
and
~(F)
is the sUbgroup reducing to the identity in
AO{F)/~(F)
A{F)/AO{F)
yare the coordinates of a minimal model for
-
is always finite; when
v(j) ~ 0
over
F.
Then
-
A (k) , and the quotient ns
A (k). ns
is isomorphic to the finite group
A
Similarly, the quotient
it has order ~ 4.
In general, the
structure of this group is determined by the special fibre of Neron' s minimal model for :es-
A
R.
over
~ (F)
The sub-group minimal model for
is a profinite p-group.
A at the origin
0 A ' using
gives the addition law for a formal group re
the subgroups in the ideal
An{F)
,,~.
A
z
Expanding the addition law on a
= -x/y
of dimension
can be identified with the points of
as a local parameter, lover
.
R.
For
n > 1
A whose coordinates lie
We then have:
(6.2.4)
.ona-
ion
6.3.
Much of the local theory simplifies when
char(k) = p
is greater than 3 .
For example, we have the following result on local torsion. Lemma 6.3.1.
Assume
v( 6) = 0
1)
If
veAl > 0
then the group
2)
If
veAl = 9
and
and
v(j) ~ 0
A(F)/AO(F)
v(p) < tcP-l)
:!i.!!.!!!!.
is isomorphic to A(F)p = (0) •
A(F)12'
;:us
16
f!22!o A(F)/A O(F)
1)
Under these assum ptions ,
has order ~ 4 2)
A
AO(F)
is a 12-di visibl e group and
1.
i-ae
0
has a minim al model of the form: curve )\ (1.1.) )
AO(F)
as
= {p = (x,y)
A has a singu larity at Let
TT'
(i,y) =
vex) < 0
(0,0)
be a root of the equati on
u
and
v(y) ~ O} . actio
0
4TT
=0
and let
E
= F(TT')
Over
•
Th'
E
we can change coord inates :
I
(poJ
x = Y
=
X/TT·
6
y/TT,9 is a1
to obtain a model for
(
p I
A with good reduct ion:
geny cl'
10: ADy point of order
p
in
A(F)
~ore mapped to the subgro up
A (E)
lewton polygo n for
over
[p];'(z )
must lie in the sUbgro up
under the coord inate change (6.3.2 ).
3
E
lincet his polygo n begins at the point >ur hypoth esis that
AO(F ).
would have
(p-l)
integr al slopes
It is there-
i
(7.2. J
Hence the ~
-3 .
When
:i'
(l,vE( p)) = (l,4·v (p» , this contra dicts
yep) < t(p-l) •
onica~ "
polyno mj (Tate
:
isogen y
I
17
1.
.t-adic Galois representations 7.1.
Let
T.t(A)
action of
be a perfect field and let
F and l
curve over
Then
F
(Reference:
Serre [19J). G
= Gal(r/F)
If
A is an elliptic
is a rational prime not equal to char(F), let
is a free
E.t-module of rank 2 which admits a continuous
Z.t-linear
G
The natural map
is always an in.1ection. geny class of
A/F.
7.2.
F
When
Consequently the G-module
=~
A(~) ~ ~/W
and
V.t(A)
depends only on the iso-
, there is a natural isomorphism
ree
When
F = Il , the action of
plex conjugation on
When
(7.1.2) is known to be an isomorphism
(Tate [30J); hence the characteristic polynomial of isogeny class of
elements, the group
A over
F.
a completely determines the
18
This important invariant may be calculated as follows..
endomorphism
of degree
TI
The curve
A
has an
q, which on the coordinates of a ·Weierstrass model is
given by
This is the Frobenius endomorphism; it is defined over
phism
~t
of
Tt(A)
t # p.
for all
Clearly
a
F
and induces a G-automor-
acts via
~t
on
Tt(A) ; some-
(
.
what deeper lie the formulae:
where
1T
+
v'
~t
=~
+
~
Det
~t
=~
0
~
-I
pla
= deg
is interpreted as an integer in
1T
teristic polynomial of The group A(F).
v
Tr
K(F)
a
1T
= 'l.
En~(A)
Conse'l.uently the charac-
is integral and independent of
t.
is precisely the kernel of the separable isogeny
(1-1T)
on whE
Conse'l.uently,
,1
(7.3.3)
Card(A(F»
= deg(l-1T) = (l-1T)(l-~) = 1
- Tr1T + 'l. .
by
,7,
One has the Archimedean ine'l.uality:
y
as well as the p-adic criterion:
Tr1T _ 0
7.4.
If
F
is a finite extension of
G = Gal(F/F)
and let
quotient
=Z .
G/I
~
(mod p)
a
A is supersingular.
~,
let
I
be the inertia subgroup of
be an arithmetic Frobenius in G which generates the
If t # p
then
a
acts on Tt(A)
I
and its characteristic
polynomial: has integral coefficients which are independent of factor: L(A/F,T)
= det(l-aT
t.
Define the local
pa
19
When A has good reduction over
n I
I
F,
I
acts trivially on Tt(A)
and
•
'S
L(A/F,T) = I - TrrrT where
TI
is the Frobenius endomorphism of
q = Card(k)
A
+
over the residue field
Card(A
places
v
2
k
and
In general,
,e-
7.5.
qT
ns
(k)).
If
F is a number field then A has good reduction at almost all finite
of
F
We may therefore define the integral ideals.
N(A) =
n Ev
V(N(Av ))
v
l.I(A) =
n
EvV(l.I(Av ))
and
v(N(A ))
v
where
.e,.
in F
,the completion of
v
by
is a prime at the place
F at
v
v.
v
and v(l.I(A)) v
are calculated
Similarly, we may define the global L-series
the Euler product: L(A/F,s) =
IT L(A/Fv,~-S)-l v
B,y
(7.3.4) this converges for Re(s)
>
~
The knowledge of
product is equivalent to the knowledge or Vt(A) isogeny invariants.
l
.J
as a
L(A/F ,s)
Gal(F/F)-module.
as an Euler Both are
s 20 8.
The arithmetic theory of complex mUltiplication
(References:
\
Serre-Tate [21J,
values
Shimura [22]).
8.1.
Let
F
be a number field, and let
A
be an elliptic curve over
complex multiplication by an order in the imaginary quadratic field
K
F
with
(8.:
3
Fix an
isomorphism:
(8.?4
(8.1.1)
By composing K
l!lL!!.
Then there is an
A with complex multiplication bY 0 ~ H with
j (A) = j
~
XA,"X .{ '0','
p!ir
~l',ii¥ cM(,t~'
X A
determil1eS1;li&"i~ije#'c14slliot 'A'o~i k
(j{A)~l(~)d&terDdl1eBthe isomorphi~i;lail'ij'brA. ()vern.
isomorphic over 'H
iff they are iSOflenoullOv&t'
it'
W~.will prove ~l4s result in the next ,8~,t1on; varieties, s,ee, Del1gne [7] and Shimura
12~]~
'l'\iocurVes are
's.naisomOX1?llic over fOT
.atld, the
ii'.
generaJ.i,zations to abelian
24
such that e(,,)
for all
"g
0
'"
and
o
w
0
= "'"
in
1
R (A,U ) •
g
1/1 :
A are define d over
Since all endomorphisms of
W in
H, any class
may be repres ented by a contin uous homomorphism
1/I:G
The Artin homomorphism
.*
-4
ab G
h:IR/R -
0*
Sine
allows us to view
1/1
as a contin uous homo-
morphism
t=
(9.2.4 )
"'0 h:IR -K*
Sinc hi 'e
which is trivia l on the princi pal id?les . If
Lemma 9.2.5.
B
= A1/I
, ~ B has comple x multi plicat ion
by
0 E!l!!. H crv
.mX A· . )i·X B=
ChUo: s.D iLlQlllOrphiSlll
Proof. .Choose
over H: +:A ...:.' B Ol
CI8lU'ly B hu 1.nq..tipllcatiOIl bT 0 over cODlpCldtioll
NoY let
a:
pr11IIe
reduce s y;
,
0
,.1 . ~.s
all
endom orphism
I fOr'
1/oI\f, endC'!"~rP!llS1!!
; .., 01; A
Of~ ~'. ~ e~l'Jl~lli~~ &ctiOl1 CD
~ s~,.~tJ~;;~~~>9VerB. ! " ; ' £) ::.;+;.~ ,~\,:.;>;:~. ~', ,.,..:(."
BO(B,lh be
+'1. Q
ii
v
be e. phce ot
at:;~d'i:et"Q~(A)
t~ the
define
P'roben 1us
0y(B)
11 y
lIJ.:~'. ~"'tled' ~ . e." Unique·ei..itht~ l:it:'O'~iBii~(Af'1fIi16h' '.'
B Where' 'both
on
be the
A.
X A
,e.na.t
ti' '13' also 'l'he v&:z;ie •
~s·
8004 Teducti4
acts on
cl(lC}
ove:rH
with C~'-')';'-':.-,-'-- ::"~--'-~'" -;~';~'/\ :~ --:?:f'::,;~:A· . -i~;': . . _ , ' _ ' , . ,:.:".\? ",.::~}:", 2:;i;;f "._ . " . "-.:,?:;; ;':-,;~'r:~~:-y:~';:~i:;,-:.:L A • ~ A curves repres ented bY. the c-'." . ,. ", . ,.,.... !Il chsse s. _. '_.. .. _ ' " , ',',".. actlY 2 F.i,!l!P,-.•"tPb!'l .." . " ," ", ,_. ".... . . .
:/;'1.-
ana
~.
curves becane isO!OOrphic over
a,
where the F-isogeDjY .:A -
-:.:-.'
_... ' '"
becom es a com-
/'_;"-
andll"~;' is6g~us';'er i{
.:1 -·B and let
•
•
(En'1l(A)8Q1) ... K
(9:1:3).
has
31
Conversely, if
2)
If
ther that Res
~
G
o
= 1
10.3. F
A and
= j(B)
j(A)
Bare isogenous over
then
A
= B~
F
~
for some
K* "Which satisD H
is a principal idHe.
= (av )
is an idele with a
v
=1
for all vIOCl,D
a(xD) also satisfies the conditions of (11.2.4) we have
(11.2.5)
I
Combining this "With (11.1.1) "We see that Qj-.curves over
(11.2.6)
H
If
G = Gal(R/H)
X gives a canonical isogeny class of D
"We have a bijective correspondence:
ISOgeny classes Of} ... , _ _...., Jt-(G,O*)Gal(H/IIl) [ (Q-curves over H {A}
11.3.
When D is
~
the construction of Ill-curves is more delicate, and in
some cases it is impossible (Shimura [25]). that Ill-curves eJP.st whenever Pii!
D 18
3 (lIlOd'4). Wheu
andt,t1,
We leave it as an exercise to check either 'by
8
or by some prime
Chapter 3
Local arithmetic
Clear
In the remaining sections of these notes we will restrict our attention to elliptic curves
A
with complex multiplication by the integers
quadratic field
K
of prime discriminant -p.
theory, the class-number then
h
of
K
is odd.
Then
p
=3
1
0 of an imaginary
(mod 4) and by genera
We shall assume further that
p > 3 ;
0* = •
groups subgr
1
Galois
are a L 12.
A classification over
12.1. and let
F
F.
Fix an invariant
= lII(j)
j
unique £
J
of a curve with complex multiplication by
0
ment If
• Recall the field diagram: then
at the
(12.1.1)
12
F = 1II{j) gives
xp .
1
~
class.. . J
L _ 12.1.2.
Let
F8 '\
ll:p. K8 '\ ... V~)
~ ,\x
K;h-l) /2
.~
F8:lt"'m II
Then
a:(h-J,ll~"
r 35
(r-p) =
(p)
Clearly 'I
Ei
Let
G(Ei)
subgroup
I
for all
p
G(E ) i
groups
I
= K
H
=
E1 E2 • .. E" 2
2
2
E1 E2 "'Eh
i
be the decomposition group of the place
E
i
in
Ga1(H/~).
all have order 2 and form a complete set of conjugates.
Cl(K)
has
~
order, there are precisely
h
Since the
elements of order
Galois group, and they form a single conjugacy class.
Hence the sub-groups
are all distinct; renumbering them we may assume that
G(E ) = . 1
unique prime
E
which divides
p
and ramifies in
H
2
E = E 1
The
2
in the G(Ei )
Then
F
has a
A similar argu-
ment with the decomposition groups at infinity gives the lemma. If
then E
E
is the prime of
= discH/F'
at the place
12.2.
E
F
which corresponds to the unique embedding
Consequently, we see that .!!:!!Z descended curve has ~ reduction
(10.3.1).
Since the discriminant
-p
of
K is odd, the construction of (11.2)
gives us a distingUished isogeny class of lIl-curves over x
p
•
F . i ; , ,--' ",
.',
\
phiSlll ,:Z/pZ _'·0 ·.over . F.~ ·we obibain squa (13.1.3)
a~
.-1
0
a(,)
~. we s 0"
'r
-
39 Conversely the Galois character of
C. When
th
P
E completely determines the F-isomorphism class C
c
=
~
p
,
the associated character
=
E:
gives the Galois action on
E:1J
P
-roots of unity in
This character is sur,j ecti ve, as
F.
equation is irreducible over
FE.
When
c= u
lIIk
cyclotomic ; hence we
we have
P
must show:
EC(p)
(lE::l) 4 = E
(13.1.4)
(.E!!) E -J c(p)
To show that
EC(p)
(or
Hom(G, (Z!p:ll:) * ) .
in
4
= E
is a power of the cyclotomic character
EC{P»
it suffices to check that it is trivial on the subgroup restricted to the larger subgroup (mod
f-P)
EC(p)
p
•
But when
is simply the reduction
of the p-adic character, p :Gal(F!H) p
which, in turn, is determined by ramified o1l.tside
AutOlll:ll: (T A(p» p
XA(p)
p
(8.3.4).
This shows that
p; to see that it il trivial onGal(F!r(~,p»
tdVialon all Frobenius elements rational prime 'i. ;; 1 (mod p).
But
Gal (F!H)
Gal(F!F(\1 »
e::,
in' this case,
A is a prime: of V
EC(p)
is un-
we must show it is R dbiding a
40
Since
C(p)
and
.-
C(p)
are Cartier dual:
.-
k + k
_ 1
'1'
(mod p-l) .
Since the two groups become isomorphic over
H :
s
Combining these identities t we see that either
k
or
if
¥
~ (mod p-l)
(mod p-l) ; the other is congruent to
To complete the proo~. we must distinguish the curves F •
is congruent to
A(p)
and
A(p) *
over
we see
or
. Note that exactly one o~ the groups
'I..
over
Hencl1t
and
is pointwise rational over the Bub-~ield M+ group o~ the real place o~ are
F.
M+-rational precisely when In
+
M
the prime
E.
p
(13.1.6) .•. - ,",",.'.'.,+ v q,(/l(A(p) . 1M .
»=
C(p)
(mod 8).
is totally ramified:
= ,
= F(~ )+ ~ixed by the decomposition
It s~~ices then to show that the points o~ p" 3
E.
=q
(J2::l) 2
We find:
r;ifP [: :: I
When
V
q
I
(6)" 9
r
ill poinh111e rat:10Iltll..
!;las ker unique real completion
ease
1\.
41
Theorem 13.2.1.
{
sign c6(A(p)) = (~) = p
Since the real embedding of
F
+1
if
p
7
(mod 8)
-1
if
p - 3
(mod 8)
-
corresponds to the homomorphism
L_-n,
1Il{J)
j I--->- j (0)
we see that. using the notation of (5.2).
over over
F:m.
FurtheI'l.ore. the curves
FE; they become isomorphic over
Hence the statement of Theorem 13.2.1' Theorem 1:>.2.2.
A(p)
2Y.!:!:
becomes isomorphic to either
A(p)
and
A(p)
'4
are not isomorphic
H. where the real plac,e of
F
is e'luivalent to the following.
Fit
A(p) =
*
t: ",
if
p
-
7
(mod 8)
if
p
-
3
(mod 8) •
;
..
A
is ramified.
42 14.
Local invariants and global torsion.
E
,
the 14.1.
~-
In the previous sections we have described the local behavior of the
F curve
A(p)
at all places of
is a finite place of over
F
v
F dividing
p
and
F which does not divide
~
On the other hand, if
p ,then
and the number of points on the reduced curve
A(p) A(p)
'r
has good reduction is determined by the
In par
Heeke character
call
It is now a simple matter to compute the local invariants of any descended
curve
B
.
v
= A(p)$
we have only to see how the invariants of A(p)
quadratic twists.
behave under
We summarize these results in the following table. Table 14.1.1
Local invariants of Kodaira S bol
B
theore
= A(p)$
T Comments:
III III*
B(Fv )/BO(Fv ) = Z/2 •
= (Z/2)2
10*
B(Fv)/BO(Fv )
1
B supersingul$r
0
I·0 '.
B ordinary _
ii
•
10 .
F
(&) = +1
P supersingul$r _ ..
B(Fv)/B~{Fv) .. B(?v)2
(~) P
Since '" -1
pro, C·
B
1
0
The sui
1 4•
vl2
3
6
{:8
dee,
I 8•
if.
II
if
+1 ber
locr'
a
f I
S'W,";-I
43 Similarly one has the following generalization of 13.1.2.
r-p
the kernel of the endomorphism
on
Let
C(p)W
denote
A(p)W , considered as a group scheme over
F. in
Theorem 14.1.2.
In particular, when A(p)W cally to
Hom (Gal (F/F), (Z!pZ) * ) •
is a Ill-curve, the group scheme
C(p)W
descends canoni-
Ill.
14.2.
As an application of our local results, we will prove the following
theorem on global torsion: Theorem 14.2.1.
1)
2)
Proof. Since
If
A is any descended curve
A(F) torsion --
A(H) torsion --
[
(Z/2Z)
if
2
splits in
(1)
if
2
is inert in
if
2
splits in
if
2
is inert in K
[ 0/20
= (Z/2Z)
2
(1)
First we will· show that the order of. A(F)torsion
A . bas bad reduction at
.e
K K
K
is at most
2 •
•
groUp' is&. direct
r:
.
of Type !Uor IU' the
pro~tj
I I
,,
The
.~b.-.sroup
, . . . ,; .
AO(F)
is a pro P-grOllP.
.e.
but ,l1O
. .'
p"torsion ,C41l exist
. deed. 117 CebOtar",'s densitT theorem we c~ find .
In ..
.t t
-1
(mod p)
where
A has good;
: -...
,..'
a prime.
..
-
a1oball7. In-
A. ot. F ",.th
jupersi~ar red~t1oil•. Sillce the num-
:.> --,-."
ber of points on the reduced curve
l~cal grO~A(F>.)
has no
A
is then t + 1
(whiCh is prime to
P-tor~io~. cons~4;u\ut~~ A(F)t~sion 1.1
p). the
isomorphic to
iii ~ll 0
(the completion of
H at
v), we obtain the following commutative diagram with exact rows
A(H)/rrA(H)
0--+
1
(17.1.1)
o -Th(H v
Here G· Gal(ii/H)
v
and
~(G,Arr)
~
~(G,A)rr
-
!Res
)/rrA(H ) v
G
v
n~
Vv
iRes
'lJ~(Gv,Arr) -l!~(Gv,A)rr
is the decomposition group
Define the 'Se1Jller group for
".' S,,(A/H)
to be
t1n#e'~1lp &lid the resUlting injection
can be
u.a84 tc
A(H)/"A(H)
bound t~e rank of A
a
0
Gal (Rv/H ) v
n~
Then S,,(A/H)
v"
is a
S,,(A/H) . ,,'
The calcu.1.ation of S" (A/B)
"mit. iiifth~ first' " ....descent. " . Define the Tate-Shafarevitch ll1'Oup W(A/H) t"1on map
I
-
t~esu.1lgroup of al(G,A,,)
'l/hose elements, under restriction, lie. in thei.mage 01'
o-
-0
-'.-
to
. '
"'--'
'.
is called
'-
be theternel of the restric-
54
Then a simple diagram chase in (17.1.1) shows that
coker a ~
U](A/a) n . Here is
the full picture:
A(p)
Gal I tha+
o
sem_ .
1 W(A/H)
n
->-0
the: .;
1
WhE
e
. g(,(cr(R))) - Ha(R))
II ,(ga(R) - a(R»
as
,€
Enindep~cl...ntot·i
wh
59
The first statement has been generalized by Deligne to a conj ecture on the con-
jugate L-series of a motive over k = K and of
i
E = T
By
(19.1.1)
f~i)(z)
the parity of
(20.1.3).
vanish simultaneously, if at all.
,,
with coefficients in
We shall show that the L-series
conjugate new-forms
whenever
k
neAl > 0 .
L(X~i) ,s)
E
[7].
In our case,
ords=lL(X~i),s)
is independent
are the Mellin transforms of
By Shimura [26], the
h
values
Furthermore, Arthaud [1] has shown
L(X~i) ,1)
L(X~i) ,1)
= 0
60
19.
The sign in the functional equation 19.1.
Let
Tr
(Reference:
A be a Ill-curve over F.
Tate [31]).
By (18.1.7) we have the factorization
L(X(i) ,s) where X is a Hecke character of K. In the next two B i=l B sections we will study the analytic behavior of the h conjugate L-series
L(A!F,s) =
Since
X depends only on the F-isogeny class of B without loss of generality, that A = A(p)d with (p,d) = 1 .
Theorem 19.1.1. A(i)(s) =
(19.2.
(19.2.
A , we may assume,
Ind'
.!!. A = A(p)d ti1h (p,d) = 1 and B = RF/uf ' the function
i
plies
Fine 1
(Pd/2w)Sr(s)L(x~i) ,s) satisfies the functional equation:
with • sign d • I
f!22!.
Since
Ix = (;:P'd), we have
Furthermore the character Hence the terms
s
i
X~i)
has type
2 and (Pd)s = Ms /
(1,0)
(2w)-sr(s)
when restricted to
* K..
=
in 17.1.1 are precisely the ex-
ponential and infinite factors in Heeke's functional equation for
L(X~i),s)
To
cOOJe
corres
complete the proof of this theorem we must Bhov that the global root number sign d.
We
! c = ,
(19 !,
Ie" Ill-c'''''
61
w
=
wv = 1
if
00
v
%pd
(19.2.4)
if
Indeed, we may choose an isomorphism
plies (19.2.2).
cOv
with
= (f-v )
so that
(19.2.3) is clear as both
Finally, in (19.2.4)
(f) -v
Xv
and
is the conductor of
v ,and ljIv
• V
vlpd.
K v
Xv'
are unramified at c
is the "canonical additive character" of
is at most quadratic when restricted to
=
the character
character of
If
vlt"
p
Xv
v
The character
it takes values in
* p(T)
1 be the quadratic character OfJII~
ljI
let
2 • 1) w = (-) P P
ov :
K
=
always restricts to the unique quadratic
r.
corresponds to the abelian extension
Theorem 19.2.5.
.*
0 v
is an element of
If remains to compute the product of the local root numbers.
Xv
v
IIlt (,d).
i
Then
Xv
= ljItd
0
which
*
lNIC.v/lllt on 0v
•
--.,.'- .. '
"....
Vp = (,r.:p).
;;
We may take
62
and
= x(p) = x
1
~
(php (p)
= ill p Xp (p) = Xp (p)
making the change of variable
b = 2a
wp
=3
Since
p
gives
1) •
= Trw
.
If
t ; p. tid
but
v
vlt
We ~ take
c = t
gives:
=
4), the Gauss sum in the numerator is equal to ilP.
(mod
Now assume
Subst.ituting into (19.2.6) and
%d
If
t
t;
2 ,then
then
Xv
$~
is trivial on
~t*
is tamely ramified and
and
This
$~(-1)
$~(-1)
=1 =
= Ce1 )
.
in (19.2.4); I claim that
(19.2.7)
Again the product on the left side is independent of the Ill-curve chosen, so we aSSume that
t
A
= A(p).
= (-) IT Xv(t). p
Then
1
= X(t) = x~(t)~(t)1f vlt
Xv(t)
~
= JfL(~):rr Xv(t) p
=
vlt
There are two cases to consider:
vlt
a)
•
63
b)
(~) p
= -1
so
(.t)
is prime in
0
I
w.t
aEJFi'2 .t
= X.t(.t)
(II,,) e2~iTra/.t .t .t
.t-l
-( I
= (-1) a=l
where the first identity follows from the theorem of Hasse-Davenport and the second follows from Gauss' s determination of the sign. Hence, for all odd d
l
we have shown
When
~/l+8~ to consider.
there are three characters of
.t = 2
divides
This can be done by hand,
and we leave the proof to the reader.
Corollary 19.2.8.
Proof.
w = (g) p • sign d .
By (19.2.1)
w
=1T w
V = wm •
V
-1
w= 1
I 2 =H p
•
W"p
(TT wv ) vi! .
•
SUbstituting in the
1
64
20.
!i-curves and modular forms 20.1.
For
N > 1
(References:
Shimur,. 123J, !24]).
let c _ 0
As a discrete SUbgroup of
PSL OR) 2
the group
(mod N)} •
ro(N)/
acts on the upper half-
plane, and the quotient may be compectified into the complex points of a projective curve
XO(N).
The curve
field
~(j(z),j(Nz»
Theorem 20.1.1. defined over
E!:22!.
XO(N)
J
O
~
with function
(Shimura [22J).
There is a non-trivial rational map
F.
Since the curves
A(p)d
without loss of generality, that Let
has a canonical model over
= J~(p2d2)
and
A(p)-pd
(p,d) = 1 .
be the Jacobian of
Let
are F:"isogenous, we ~ assume, A = A(p)
o = XO(p2d 2).
X
P t--+ [pHi-]
over
and
B = ~/Ill(A) •
Since we have canonical
maps:
Xa'"- J O
d
III
65
tyists all satisfy functional equations of the appropriate type (19.1.1), it follows
f~i)(z)
from a result of Weil [34] that The character of
rO/r
l
is a ney form 0:1' weight
is trivial in this case, and
f~i)(z)
2
for
rl ( p
2
i).
has a Fourier ex-
pansion:
(20.1.4)
where
q = e
2triz
~himura [23]).
By a theorem of Shimura [24], the abelian variety
canonical factor
Since by
B and
Toyer
B of dimension O
over
III
"0
has a
with
III ~ have complex multiplication
K, they are jQ-isogel\oUB (Shimura [25]).
Qu.estion 20.1.5:
This isogeny exhibits
as
B
"0(p2d2) •
Assume
The ne;, forms
wh1cb. ~e defiDed Over the .:".",
h
B have the same L-series C1'ler O
a rational quotient of
20.2.
generate the field
The Fourier coefficients of
d
=1
Is the abelian variety
B
= IT/rI(p)
f~i)(z) co;r~sPondto~l"'rphiC ditfer8nti~s 011 !tq normal closurli 01' iot: in 1ft and are etBenfOl:omS .for the
";.-~.J'2)"'\X*lr!2)2 eotD the cCllllpletioD
I
,
• I
A(P) • (a.B)
ll¥, (11.3.1)
P. A(H)
generates a biquadratic extension
R
H and we ~ choose
For
c' :' , "'.",i,'",
(H* /U*2)2
TI(U*/H*2)2. -v ..... v:
Let 0 v
Bo1 f
-
l 73 shows toot
ImA
when
vf2p.
If
v
I
2
0/20
v
then the image of
containing those elements of the form
(a,a)
A v with
is the subgroup of
aol+40
v
and
(Brumer and Kramer [4]).
Now assume
(a,a)
in
v
H
previous lemma and the fact that valuation at all places is
~
w of
not diViding
K
K,
is unrsmified over p
a
and
By the
v.
for all
A
a
have even
Since the class-number of'
K
we may write
a
=(_l)a(,r.:p)b
a =: with
a, b, c, d
onto
(H/H ) v
in
Z/2Z
This subgroup of
* *2 2 , where v has order
We nov
is any prime of
4, we see S2{A)
.a~lltld~18 a (lOPY of
(mod K*2 )
(_l)c(,r.:p)d
A(H)2 ..
study the
"..
H
(K*/K*2)2 restricts isomorphically dividing
has r&llk
0
0/20 i;y' (16.·2~S)
I:;
or and
p.
Since
lover
8
L
with
But
SO
Let 'L .. K(ll) p
"-.-
D
V
0/20
(11.
A .. A(p)·.
Im A ..
01;; 1
.
and
£!:.! '2
mod ! ! . }
..--
----------------_.~'-------
74 (22.2.2)
where
€=€
Vp
is the cyclotomic character defined in
section, we decompose
(13.1.3).
D only into eigenspaces for
characters of the full group Aut(L)
Notice that in this
Gal(L/K); later we will use the II
= Ga1(L/~) (lJcl.)
Theorem 22.2.3.
4
S (A) "D
A = A(p)
If
1T
n
Proof.
(13.1.2), Kummer theory gives an isomorphism:
Since
s
(22.2.4) 1 * * ~(Ga1(H/H),A ) " (H(v ) /H(V ) p) P
'1T
3 1 (po-T)
3 1
* (~) " (H(v )*/H(v ) p)
P
P
H(V )*/H(v )*P P P
where the right hand side denotes the sUb-space of acts via the character as follows:
for
P
£
A(H)
1?f = e:¥
p
of
(0-:;0/) ,
•
l!
the image of
(~1)-, "..
'(1 + ..P:jol'o 1(1 + 'It O:)p) v~, V"': - ',-'v.;"·--':
-
-
A may be described
Then R generates a so that
A(P)
,,'V' , .
'C';,·
H
=a
.,
~",:'1i.7.!iQ'"tO.'t:AAlluWbup "
'" .."
vip
A(H) •
p
.
"_' c.-,·-:, :'.,-, -.:', :-0;: •. :.:_,~_-., . -_:: .:: . -.".:;.:
Ga3;CL~t»/L) {;t) . h
SinCe
Np)
j;rhie.l
byl)••
c~ntabls
claBs-tield theory turn:l.Bhe,
( (I ci(] 1e
-
77 Let (mod
u" 1
E.)
(mod
j
E. +1 ) with a
t
0
be a generator of this eigenspace, and write (mod
E.)
I claim that
be an unramified p-extension of
in
Ct(L) (p-i)
b
(Z/p~)
in
*
Hence
1)
If
would
= Gal(L/IIl)
(mod
Corollary 22.3.5.
L( Pill)
< p+l ; otherwise
L which would correspond to a non-trivial element
But for any
p
j
u = 1 + art
j
E:j+l )
= i , which completes the proof of
p" 7
(mod 8)
~
Bpi?
t
0
2) •
(mod p)
= (0)
S,,Not... that. C/Tp'is an Oort-Tate lltouP scheme", G"",b of Bp . such that
a-b- p.
must have, ord(a)and ord(b) ,that .ord(h) .• 1).
Since
Since
~ere:,.a,b are elements
Ga,l;' has its sections rationsl over Tp
both dinsible by. p ... 1· (ord is ,nor:msliZed so
ord(p). 2(p-l)
this leaves
3
possibilities for
.';~d~{~~i~?~Q~li,H~"~ .~,~ell~~'fre1;ing .. ;. .\ ;'. "_'''''.. ,,::,.!_:.' ·:c· .. · .'.' .. :.. .> . . ' .... ~:'f.
Let
(A*)(j)A*P/A*P P P P
Theorem 1 of Roberts' Thesis ([ 4); see also. [3. Prop. 9.3J
fiesI1(T 'C) p
:/:~_.
();*)\J)
be a non-negative integer and let
':('~.
II
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _iiiiiiiiiiiiiiiii'''iii'''iiTiii''ii'·'ii'.'iiiiiiiiiSFiiiiiiRiiii
,l-.. .
...
89
~(T p .C)
in ~(T .~ ). p
p
We define
ffl(s.c) = ~(T.C)G .
(25.2.1)
Theorem 25.2.2.
If P is odd p+1
and ,," 1
Proof.
mod"
2
)
From (25.1.4) one obtains that the square
~(s.c) - - ihs .c)
1
i1(s.~ p ) is Cartesian. where
i1 (S ._)
=
~ (T ._) G.
(
i1(Sp .~p ) By Kummer theory
Taking invariants gives
as
pia
pr1llla to
laj • S1llIilarly V(O)a. tI(ol 'ilJldV(p)a • tI(Pl).
g1ves tbe theorem. ',:', ,,", ~,.
Co~ll!l7 25.2.3. l! p 1s
'
..
an odd.. rml1lH"~!t! 1t
'p
=1 .(lIIQd Ii)
1t.p il3
(mod 4)
This
90
!1:22!.
~(S'~J
tion,
be the units of
is non-t rivial for
E ~p
.
'+1
(mod ,,2
i > p+l
-
lIl( ~p)
The eigens pace of
is one-di mensi onal; since
u " 1 + a,,2
such
= U/UP . i
the charac t er
U/UP( i)
U = A*
Let
)
where
a
t
U/U P
on which
C!(A)p = (0) Gal(lIl(~p)/lIl)
i=1, 2,4, 6, ... , p-3.
by as sumpacts via
that
,
grouI
In each case
p
is regula r it can be genera ted by a unit
to ti..-:
0
(mod" )
T ,
is theref ore the dimen sion of
2
Since
(compa re (22.3 .4».
The number of
genouS
ffl(s,c) .
that
25.3.
We shall apply the above compu tation to retrie ve (part of) a result of
Faddee v [1 J•
Let
K
= lIl( ~p ) = lIl(,,)
plex multip licatio n by
K.
over
",
and let
J /K
be an abelia n variet y with com-
posses sing a non-t rivial point of order
p
is an odd regula r prime.
(ii)
J
achiev es good reduct ion everyw here over
Examp les of abelia n variet ies with multi plicat ion by p" 1
(mod 3)
Define the 'r-flelm e:r numbe r of. J
Theorem 2'1' ~.l.
"
p
1Ihere
Over
in
.1:
as
P
satisfY ing p
(ii)
are
where the
Th
has a "tame" quotie nt (Gross "';Roh rlich
by
Under the above hYpOt heses:
. -t:
Proof.
ration al
T.
and, more gener ally, for those primes
Jacobi an of the Ferma t curve of expone nt
[2] )•
"
Suppos e furthe r that:
(i)
known for
quence
T' 'We haVe 'the follow ing short exact
2
3
4.
a
91
mp-
that
is an isogeny of abelian schemes) and therefore
ia
group scheme over
1T
T
of order
non-trivial point of order to the constant group f
T ,
ker
1T
_
1T
p
= deg
n.
is a finite flat
11"
By the assUMption that
it follows that
,
ker
~
-ker
J(K)
is isomorphic over
is either isomorphic to
ker n
T[l/p]
By the discussion in (26.1) we may conclude that, over
7ljp
Zip,
or
C,
~
p
.
Since
ker
1T
genous to its dual, it cannot be either etale or of multiplicative type. that
has a
is iso-
It follows
C , and the exact se~uence (26.3.2) gives the se-
is isomorphic to
quence: o
-T
in flat cohomology. as
p
J(T)/1rJ(T)
The
Jil-(T,C) -
-T
G.
-T
0
invariants of this se~uence remain exact,
G = Gal(T!S)
is prime to the order of
W(J!T) n
A s:jmilar argument shows that
J(T)!nJ(T)G
= J(K)!~J(K)
W(J!T)G = ~
The asserted formula then follows from
W
(J!K)
~
•
d~p:D: J(K)!~J(K)
=1
+
d~(K)&AK
.
References:
k R. in "Iilvariants of divisor classes··tor the ·c:urves x' (l-X) " y fieid. Tritdy Math.:- (in Russian) Inst. Steklov 64 ....adic etelotomic
Faddeev,. D.
. e.n
1(;
(1961), 284-293. 2.
Gross, B. H. and Rohrlieh, D.E. SOllle res1lJ.tJl anthe·M!iI'4eJ.J.,-wehgroup of the il"aeobian af the Fermat curve. !nv. Math. 44 q.9'(8) ,.201-224.
3. Mazur~ B. and Roberts, L. -201-234. Roberts, L. On the flat eohomolollY of finite group scihemes~ TheSis. (1968).
Harvard
26. 1.
Arthau d, N.
Prepr int.
S}
23·
Berwic k, W. E. H. Modul ar invari ants expre ssible in terms of quadra tic and cubic irrati onali ties. Proc. London Math. Soc. (2)
3.
Birch, B. J. Dioph antive analys is and modul ar functi ons. quium on Algeb raic Geome try (1968) , 35-42.
4.
Brume r, A. and Krame r, K.
(1977) , 715-74 3.
6.
22.
On Birch and SWinn erton-D yer's conjec ture for ellipt ic curves with
28 (1927) , 53-69.
5.
S,
Biblio graphy .
comple x multip licatio n II. 2.
21.
The rank of ellipt ic curves .
Duke Math. J. (4) 44,
On Epste in's zeta-f unctio n.
Crelle J. 227 (1967 ),
Chowl a, S. ana Selber g, A. 86-ll0 .
24.
~}
25·
:1
Proc. Bombay Collo-
26. 27.
s
28.
T
29.
~
30.
[
31.
,~
32.
1
33.
I
Delign e, P. Courbe s ellipt iques : formu laire. Modul ar functi ons of one variable (Antwe rp IV). Lectur e Notes in Math. 476 (1975
), 53-73.
7.
Delign e, P. Valeu rs de foncti ons L et p~riodes d' int~gr ales. in Pure Math 33 (1979) , part 2, 313-34 6.
8.
Deurin g, M. Die zetai'U nktion einer algebr aische n Kurve von Gesch lechte Eins. I - IV. Gott. Nach. (1953, 1955-1 957).
9.
Gross, B. and Kobli tz, N.
109 (1979 ), 569-58 1.
Gauss sums and the p-adic r-func tion.
10.
Gross , B.
ll.
~,
12.
Lang, S.
13:"
Li~Z::r.G. {:=~~~t~~
14.
S.
1faZ1U','~;
Ill-curv es and p-adic L-fun ctions . Cyclot omic fields . Ellip tic
In prepa ration .
lIMtBo n-WeB le;y. {:l973h..
ModUlar. curves and the
~ .',"',
Annals Math.
Spr1ll ger-Ve rlag (1978) .
~f'wlctlons.
, ;v,,~,,' ,(~~).~~~(l6~,>
Froc. Symp.
