Ka- I
A RESULT
ON MODULAR FORMS IN C H A R A C T E R I S T I C
Nicholas
p
M. Katz
ABSTRACT
e = q ~d
The action of the derivation
modular
forms in characteristic
in the S e r r e / S w i n n e r t o n - D y e r
this note, we extend
the basic
results
P > 5 and level one,
prime-to-p
level.
!.
of modular forms
We fix an a l g e b r a i c a l l y
p > 0, an integer
of u n i t y
~ ~ K.
K-algebras
N > 3
this action,
p
in characteristic
closed
prime
N
about
to a r b i t r a r y
to
field
p,
The moduli problem
with level
represented
is one of the fundamental
theory of mod p modular forms.
known for
Review
p
on the q-expansions
structure
tools
In
already
and a r b i t r a r y
p
K
of characteristic
and a primitive
"elliptic
~
of
curves
N'th root
E
of determinant
over
{"
is
by
(Euniv i ~univ)
MN
with
MN
a smooth affine
the invertible
sheaf on
irreducible
curve over
MN
I
= ~, flEuniv/M N
the graded
level
N
ring
R~
modular forms over
O
k~
nowhere-vanishing
'
of (not n e c e s s a r i l y h o l o m o r p h i c
Given a K - a l g e b r a
In terms of
K.
B,
invariant
K
at the cusps)
is
H O ( M N , @ f k)
a test object
differential
(E,~)
~
on
over
E,
B,
and a
any element
54
Ka-2
f e RN
(not n e c e s s a r i l y homogeneous)
and
is determined
f
Over
by all of its values
B = K((ql/N)),
"canonical"
differential
"is"
from
aso
dt/t
~m ).
of determinant
K((ql/n)),
we have
C0can
(cf.
e B,
[2]).
(viewing Tare(q)
on Tare(q),
f(E,~,~)
the Tate curve Tare(q)
By evaluating
~
we obtain
corresponding
has a value
as
with
~m/q ~,
at the level
N
~can
structures
all of w h i c h are defined
the q-expansions
of elements
its
over
f e R N at the
cusps:
dfn
fm0(q)
A homogeneous
k
f e R kN
element
and to be a cusp
q I/NK[[ql/N]]
R~,holo
of
follows.
dual to
RN,
invariant
of itself.
A(E,~,~)
defines
B,
K[[ql/N]],
lie in
a subring
ideal in
of
RN,holo.
is defined m o d u l a r l y as
let
ri c HI(E,OE )
The p'th power e n d o m o r p h i s m
= A(E,~,~)-,~
HI(E,OE ),
be the basis
x -~ x p
w h i c h must carry
N
in
of
r;
to
HI(E,OE ),
w h i c h is the value of
of the Hasse
A o(q ) = l
For each level
olo
is said
So we can write
c B,
the q-expansions
q-expansion
over
an e n d o r m o r p h i s m
~P
for some
lie in
forms constitute
A c R
by its weight
f c R kN
and the cusp forms are a graded
(E,~,(~)
]
~ c H0(E,CE/B).
a multiple
A form
form if all of its q-expansions
Given
induces
determined
if all of its q-expansions
The holomorphic
The Hasse
All
is u n i q u e l y
and by any one of its q-expansions.
to be holomorphic
OE
f ( T a t e ( q ) , ~ c a n , ~ 0) e K((ql/N))
in
invariant
K((q~/N))
structure
so
A
on
(E,~,~).
are i d e n t i c a l l y
]:
for each s o.
on Tare(q),
the c o r r e s p o n d i n g
ring h o m o m o r p h i s m s
whose kernels are p r e c i s e l y
the principal
ideals
(A-I)R~
and
55
(A-l)R~,holo
r e s p e c t i v e l y ([4],
k
f ~ RN
A form
not d i v i s i b l e b y
f'
k~
RN
q-expansion
I!.
A
that
in
f
R~,
([4],
or equivalently,
which,
k
if it is
if there is no form
at some cusp, has
the same
does.
S t a t m e n t of the theorem,
The f o l l o w i n g
[5]).
is said to be of exact f i l t r a t i o n
k' ( k
with
~-3
and its c o r o l l a r i e s
theorem is due to Serre and S w i n n e r t o n - D y e r
[5]) in c h a r a c t e r i s t i c
P > 5,
and level
N = I.
Theorem.
(I)
There exists a d e r i v a t i o n
increases degrees by
is
q~
d
p + I,
which
effect u p o n each q - e x p a n s i o n
:
(A0f)
(q) = q d_ (fso(q)) for each
So
(2)
divide
and whose
A 0 : R ~ ~ RN+P+~
k,
particular
(3)
If
then
f
dq
R kN
c
A0f
has exact f i l t r a t i o n
has exact f i l t r a t i o n
k,
S O
and
k + p +
p
~,
does not
and in
A 0 f ~ O.
If
f c R~ k
and
A 0 f = 0,
then
f = gP
for a unique
k
g c RN .
Some C o r o l l a r i e s
(])
The o p e r a t o r
A0
to the ideal of cusp forms.
(2)
If
I ~ k ~ p - 2,
then
g
f
then
injective.
(4)
If
the subring of h o l o m o r p h i c
f
has exact f i l t r a t i o n
of w e i g h t
] ~ k ~ p - 2,
f
the map
(For if
is h o l o m o r p h i c
g
then
g
vanishes
f
(For if
hence
of w e i g h t
has exact f i l t r a t i o n
at one cusp,
f = Ag,
g = 0.)
is
the theorem.)
is n o n - z e r o and h o l o m o r p h i c
then
constant; as
k.
k
~ Rk+P+1
A 0 :RN,holo
N,holo
and vanishes at some cusp,
f = Ag,
of weight
k - (p-l) ~ O,
(This follows from (2) above and
If
forms
(Look at q - e x p a n s i o n s . )
is n o n - z e r o and holomorphic,
is h o l o m o r p h i c
(3)
maps
p - l,
p - ~.
of w e i g h t O, hence
it must be zero.)
Ka-4
56
(5)
(determination
of
Ker(Ae)).
then we can uniquely write
f = Ar.g p
f ~ R kN
If
with
0 ~ r < p - I,
r + k ~ O(mod p), and g ~ R~ with pf + r(p-]) = k.
by induction
If
r ~ O,
the
theorem
on
r,
then
the case
k @ O(p),
f = Ah
for
but
resp.
invariant
unicity
!I!.
of
above,
Aef = 0.
we h a v e
if
f
(This is proven
(3) of the theorem.
Hence by part (2) of
Because
Aeh = O,
and
is holomorphic
by a subgroup of
and
h
(resp.
SL2(Z/NZ)),
f
h
has lower r.)
a cusp form,
so i s
g
(by
g).
Beginning
of the proof:
The a b s o l u t e
Frobenius
F-linear
back
In (5)
being part
k+]-p
h ~ RN
some
h a v e t h e same q - e x p a n s i o n s ,
(6)
r = 0
Aef = 0,
has
endomorphism of
(F)
Euniv
of
its finite flat rank
endomorphism
p
obtained
subgroup
is the relative Frobenius
1
F
of
MN
scheme
induces
as follows.
by dividing
an
The p u l l -
Euniv
Ker Fr
part
by
where
~(F)
> ~univ
Fr:Euniv
*
e a n d A~, a n d p r o v i n g
H 1 (E
/M ~
DR u n i v " N / '
is
Euniv
defining
morphism.
The desired map is
Fr
(F)
F r : HDR(Euniv /MN)
>4R(Euniv/MN)
~ R ( E u n i v / M N ) (F)
Lemma
of
].
The image
U
of
Fr
is a locally free submodule
H~(Eunl~n"v/M-)N
of rank one, with the quotient
H~R(Euniv/MN)/U
,Hasse
locally free of rank one.
The open set M N
C M N where A
is
invertible
filtration,
Proof.
is the largest open set over which
i.e., where
~@U
Because F~ k i l l s
through the quotient
>
U
splits
R(Euniv/MN).
HO(Euniv , ~Euniv/MN )(F)
H](Euniv,O) (F),
inclusion map in the "conjugate
the Hodge
where it induces
filtration"
it
factors
the
short exact sequence
(1)
Ke-~
57
(cf [~], 2.3)
0 -->
HI
(F)
This proves
filtration,
~, rl
H~R
<~,TI>DR = ]
~,
--> HO(Euniv ,
the first part of the lemma.
the Hodge
of
nl
Fr
-->4R(Euniv/MN)
(Euniv,@)
Then
to the Hodge
~
and so the matrix
To see where
we can w o r k locally on
adapted
projects
of
Fr
M N.
filtration,
to a basis
on
*
R
4
Euniv/MN)
of
U
- - > O.
splits
Choose a basis
and satisfying
HI(E,OE )
is (remembering
dual
to
Fr(~ (
~) )
*
= 0)
(0 A
where
by
A
is the value of the Hasse
Be + A~,
span
H~R
and the condition
is p r e c i s e l y
Remark.
and
that
According
B
A
invariant.
that
e
and
Thus
Be + A~
zero.
This will be crucial later.
(Compare [ 2 ] , A1.4.)
w h i c h for each integer
Over
e
M~a s s e ,
k > I
of
RN[I/A]
induces a d e c o m p o s i t i o n
. . . ~ U®k
The G a u s s - M a n i n connection
]
k > I
the K o d a i r a - S p e n c e r
I
a connection
isomorphism
£
as follows.
we have the d e c o m p o s i t i o n
Symmk~R ~ _m®k(~) (e®k-1
( ~ ) (~U)
_
Using
the func-
w h i c h occur in the above m a t r i x have no common
We can now define a d e r i v a t i o n
for each
Q.E.D.
to the first part of the lemma,
A
is spanned
together
be invertible.
tions
induces,
U
~
([2], A.].3.]7)
HN/K
Ka-
58
6
we can define a mapping of sheaves
®k
®k+ 2
as the composite
> Sym:L
=
--
DR
(~
--
Iil
"'"
f~MR
KS
-
Pr 1
__} ~®k+ 2
Passing to global sections over
MHasse
-N
'
we obtain, for
k > I,
a map
0,~Hasse
:H ~ N
"--
e "
Lemma 2.
Proof.
co®2
over
Hasse ,,@k+2 )
> H0(mN
,m
"
The effect of
@
upon q-expansions is
d
qT@"
Consider Tate(q) with its canonical differential
k((ql/N))
Under the Kodaira-Spencer isomorphism,
can
®can
corresponds
q ~d
.
C
v(q ~ ) ( ~ c a n ) .
~O,~Hasse~
_.~
, _®k),
on (Tate(q), some ~0)
v(f
dq/q,
the dual derivation to which is
By the explicit calculations of ([2], A.2.2.7),
spanned by
f
to
U
Thus given an element
its local expression as a section of
is
is
_
®k
f 0(q).~ @kcan. Thus
®k
dq
q
®k
®2
~can
®k )
V(q ~ - - ~ )f~0
((q)'~can)
c~0 (q)'ecan =
d
(q)'~can)
: v(q T4)(f%
d
d
®k+ 2
® k + I .V( q -d-~)(~can) '
: q d-q (f~0 (q))" ~can + k~f~0(q)" can
Because
~ (q ~q)
(~can)
lies in
U,
it follows from
59
the d e f i n i t i o n of
e
Ka-7
d
(ef)~o(q) = q d-q
(f~0 (q))
that we have
.
Q.E.D.
L e m m a 3.
For
k ~
],
k
Ae:R N
there is a unique map
~k+p+1
> mN
such that the d i a g r a m b e l o w commutes
HO( 4 a s s e , )~_ k
9_~_> re'O".
Hassek~N ,£®k+2~
U
RNk = H0(MN, @k)
Proof.
basis of
A g a i n we w o r k l o c a l l y on
by the K o d a i r a - S p e n c e r
DerNN/K
= HO(MN,fk+P+?)
> ~k+p+l
~N
A0
£, ~ the local basis
dual to
~,
M N.
I
of
~MN/K
isomorphism,
and
×A > n"O'kmN~asse,~®k+ p+ ] )
D
~' = V ( D ) @
Let
@
be a local
corresponding
to
the local basis
~ ~R"
Then
of
< @ ' ~ ' > D R = I,
(this c h a r a c t e r i z e s
D),
so that
@ and
m'
form a basis of
]
HDR , adapted to the Hodge filtration, in terms of w h i c h the
m a t r i x of
Fr
is
0B)
(0 A
with
A = A(E,e).
w h i c h is dual to
satisfies
Let
e.
u e U
Then
< e ' ~ ' > D R = I,
u
be the basis of
U
is p r o p o r t i o n a l
to
~asse
MN
over
Be + Ae',
and
k
f e RN,
and
so that
B
l
u = K ~ + e
In terms of all this, we will compute
9f
show that it has at worst a single power of
Locally,
f
is the section
f]-@
®k
of
~
A
®k
,
for
in its denominator.
with
fl
v(f~ ~k) = v(D)(f1~®k).~
= V(D)(flco ®k) .@®2
®k+ 2
®k+
= D(f] ).co
+ kf1@
1 .co,
B
= D ( f I) ®k+2 + kflco®k+1(u _ K d~)
holomorphic.
60
Ka-8
-kf
: (D(fl)
Thus
of
from the d e f i n i t i o n
~(f)
B ®k+2
• K)
1
of
e
it follows
We can now conclude
RN
But as
(e.g.,
R~
A),
Conclusion
A~(f'aPr)
AP r
for
of the proof:
f e R kN
(as section of
Rather
Locally
Ok
,
R~
of positive
of positive
extends u n i q u e l y
for
to all of
r >> 0
k
for
A
in
and
R~,
f e RN
A~(f)
This means
that
that at some zero of
®k)
than
must be invertible
A
f
A,
does
[3] that
A
at some zero
we will not make use of this fact.)
~
of
~.
Then
f
becomes
is given by
~k) = (kD(fl)_kBfl)
k
is not divisible
is invertible
at all zeros of
]).
x s MN
Thus if
f
we pick a basis
A~(f~Suppose now that
i.e.,
k.
(In fact, we know by Igusa
so in fact
MN,
(2) and (3)
(as section of
surprisingly,
on
Parts
has exact f i l t r a t i o n
®p-l).
has simple zeros,
f].~
of
Ae(f)
has a lower order zero
A.
Ae
= (AD(fl)-kfiB)~ ®k+p+1
is not divisible by
of
on elements
Up
valid.
Suppose
f
(1) of the theorem.
formula
The local expression
Ae(f)
Q.E.D.
has units which are h o m o g e n e o u s
Aef
IV.
A
the d e r i v a t i o n
by the explicit
remains
that the local expression
- kf I ~--) ® k + 2
the proof of Part
to now, we have only defined
degree
"u
is
0(f) = (D(f~)
degree.
®!<+I
+ kfl¢
A
is a zero of
by
p.
Recall
that
B
(of the remark following Lemma
A
where
ordx(f])
< Ordx(A),
61
Ka-9
we easily compute
ordx(AD(f l)-kBf I) = ordx(f I) < ordx(A).
This proves
Part (2) of the theorem.
To prove Part (3), let
expression
for
Ae(f)
f c R pk
N
have
D(fi) = 0.
Because
field of characteristic
say
fl = (gl)p"
section of
Thus
_~kP,
(necessarily unique)
g's
= 0=
The local
gives
An(f I ) ®k+p+1
and hence
A@(f)
p,
MN
= 0
is a smooth curve over a perfect
this implies
fl ~®kp
that
= (g1~®k) p,
fl
is a pth power•
so that
f,
as
is, locally on
section
g
M N,
the pth power of a
®k
of ci~
By unicity, these local
patch together.
Q.E.D.
REFERE NCE S
171]
N. Katz, Algebraic solutions
Inventiones Math. 18 (1972),
[2]
• P-adic properties
of modular schemes and modular
forms, Proceedings of the 1972 Antwerp International Summer
School on Modular Functions, Springer Lecture Notes in
Mathematics 350 (1973)• 70-189.
[3]
J. Igusa, Class number of a definite quaternion with prime
discriminant, Proe. Nat'l. Acad. Sci. 44 (1958)• 312-314.
[4]
J. -P. Serre• Congruences et formes modulaires (d'apres
H. P. F. Swinnerton-Dyer)• Expos@ 416, S4minaire N. Bourbaki,
1971/72 , Springer Lecture Notes in Mathematics 317
(1973), 319-338.
of differential
1-118.
equations,
H. P. F. Swinnerton-Dyer, On ~-adic representations and congruences for coefficients of modular forms, Proceedings of
the 1972 Antwerp International Su~mmer School on Modular
Forms• Springer Lecture Notes in Mathematics 350
(1973), 1-55.
N. Katz
Departement of Mathematics
Fine Hall
Princeton University
Princeton, N.J. 08540