The Hasse invariant A c R is defined modularly as olo follows. Given (E,~,(~) over B, let r i c HI(E,OE ) be the basis. ] dual to ~ c H0(E,CE/B). The p'th power ...
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
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.
®!
