Internat. J. Math. & Math. Sci.
VOL. [3 NO. 4 (1990) 661-668
661
MIXED AUTOMORPHIC FORMS AND DIFFERENTIAL EQUATIONS
MIN HO LEE
Department of Mathematics
University of Northern Iowa
Cedar Falls, Iowa 50614
(Received August 29, 1989 and in revised form April 26, 1990)
We construct mixed automorphic forms associated to a certain class of non-
ABSTRACT.
We also establish an isomorphism
homogeneous linear ordinary differential equations.
between the space of mixed automorphic forms of the second kind modulo exact forms
nd
a certain parabolic cohomology explicitly in terms of the periods of mixed auto-
morphic forms.
KEY WORDS AND PHRASES.
Mixed automorphic forms, differential equations, parabolic
cohomology.
1980 AMS SUBJECT CLASSIFICATION CODES.
0.
IIFI2, IIF75.
INTRODUCTION.
In this paper we shall consider mixed automorphic forms associated
to certain
differential equations and obtain results similar to the ones in [8].
In [8] P. Stiller considered a certain class of linear ordinary differential
equations with regular singular points on a smooth algebraic curve
X.
To each non-
homogeneous differential equation determined by a homogeneous linear differential
equation and an element
Z
in the function field
generalized automorphic form of the second kind.
K(X)
of
X
he associated a
As a generalized Eichler-Shimura
isomorphism he established an isomorphism between generalized automorphic forms of
the second kind modulo exact forms and a certain parabolic cohomology group.
He
determined the isomorphism explicitly in terms of the periods of the automorphic
forms.
Mixed automorphic forms arise naturally as holomorphic forms on elliptic varieties (see
I
of this paper, [2], [3], [4] and [5] for details) and they include
classical automorphic forms as a special case.
The main purpose of this paper is to
associate a mixed automorphic form to each nonhomogeneous differential equation
determined by an element
Z
in
K(X)
and establish an isomorphism similar to the
one in [8] described above. We shall also extend the Stiller’s explicit description
of the isomorphism for classical automorphic forms to the case of mixed automorphic
forms.
To be more precise, we shall determine the isomorphism in terms of the
periods of mixed automorphic forms.
I would like to thank the referee for various helpful suggestions.
MIN HO LEE
662
I.
MIXED AUTOMORPHIC FORMS
PSL(2,R)
F
Let
SL(2,Z)
:[’i
SL(2,Z).
I’
Then both
Let
transformations.
F
and
v:H
act on
2
H
v(yz)
F
c(F I) has finite index in
2
the Poincar upper half plane H by
a(y)v(z)
]’I
and that the
DEFINITION I.I.
Let
F
y
for all
Fl-cusps
k, m
and
H.
z
F
are parabolic
2
F2-cusps.
map to
k
be nonnegative integers with
(k,m)
automorphic form of type
linear
be a holomorphic map such that
We shall assume that the inverse images of parabolic subgroups of
subgroups of
Let
be a torsion free Fuchsian group of the first kind.
be a homomorphism such that
C
f:H
is a meromorphic function
A mixed
even.
which satis-
fies the following conditions:
iz+d l)k (c2v (z)+d2)mf (z),
al bl
c F
(y)
Y
(c
f(yz)
(i)
w|ere
(ii)
f
We denote by
Let
X
d
2
f
.
Sk,m(F1’v’)
2
2
f
is holomorphic and
vanishes at the
:F
Suppose that
SL(2,Z)
:E ---X
representation for an elliptic fibration
m+1
associ-
is the monodromy
v:H
and that
lifting of the period map (see [2], [3], [4] for details)
variety of dimension
(k,m)
the set of all mixed cusp forms of type
FI\HU {Fl-cusps}.
71-
(k,m) (see [2], [3], [4], [5]).
is called a mixed cusp form of type
f
F 1, v,
ated to
c
l-cusps.
is meromorphic at the
In he above definition if
cusps, then
a2b 2
d
c
Let
associated to the elliptic fibration
H is the
E
m
be the elliptic
X (see [3],
:E
[7]).
PROPOSITION 1.2
There is a canonical isomorphism
$2, m (F i’ v,()
PROOF
2.
H
O(Em,m+l)
See [3, Theorem 3.2].
DIFFERENTIAL EQUATIONS AND MONODROMY.
In this section we shall describe some of the results given in [8].
a proper smooth connected curve over
differential equation on
Anf
K(X)
for
i
X
Let
be
n-th order linear homogeneous
dnf
n
+ P
d
n-I
f
n-ldx n-I
+
+
df
PI--x + PO f
l,...,n-l.
We shall assume that
0,
K(X)
is a nonconstant element of the function field
x
are in
Consider an
X
dx
where
C.
A
n
of
X
and
P’s
i
has regular singular
points.
Let
U
X
be a Zariski open set on which the functions
0
i’’’’’
By analytic continuation of the solutions
y
are all regular
be holomorphic solutions of
is a good local parameter.
Let
n
which form a basis for the space of solutions near a fixed point x e U.
and also x
Anf
Pi
l(U,x0),
we obtain a representation
representation.
I’’’"
m
0:l(U,x O)
n
0
around each closed path
---GL(2,C)
called a monodromy
MIXED AUTOMORPHIC FORMS AND DIFFERENTIAL EQUATIONS
For
K(X)
Z
Anf
consider the nonhomogeneous equation
Z
U so that
shrink the Zariski open set
Z.
is also regular on
663
If necessary,
The solutions of
U.
this equation are given by
W
where
j
x
W
dx + c.)
z
W
0
z
Ai
is the determinant
n
C is a
n-th column, and c.
’’’’’’t
i-th row and the
by deleting the
l(U’x O)
ay
(al,...,
My
and
a
n
around a closed
a
n n
f
around
.
,
If
"r
we have
ayT
a
+
+
ala
is called the period of
GL(n,C), then
0()
hence the vector
f
is
f +
The vector
I(U’Xo)
i+n.z Z
Suppose that the analytic continuation of a solution
constant.
Y
|
i=l
is the Wronskian of the solutions
obtained from
path
(-1)
x
n
f
a +aM;
determines a cohomology class in the group cohomology
H1 (t (U’Xo)’Vx O)
with respect to the monodromy representation.
Anf
solutions of
Here
V
x
U,
0 at x
0
The cohomology class in
THEOREM 2.1.
denotes the space of local
0
HI (I (U’Xo)’VxO)
Z’
there is
K(X)
such that
AnZ
Z.
3.163.
X be the set of singular points of
S
X
monodromy and let
0
X
that are locally exact on
is exact, i.e., if and only if
Z
is trivial if and only if
See [8, Theorem
PROOF.
Let
K(X)
Z
determined by
S.
X.
We denote by
An
LE(A n)
with nontrivial local
the set of elements
K(X)
Z
Let
E(I (Xo,XO) ’Vx0
be the subgroup of
Hl(l(Xo,Xo),Vxo
consisting of the cocycles corresponding to the locally exact elements
Z
K(X).
Thus by Theorem 2.1 we have
E(nl(Xo,X0),Vxo
3.
X
LE(An)/{Anz
IZ’
K(X)}.
MIXED AUTIORPHIC FORMS ASSOCIATED TO DIFFERENTIAL EQUATIONS.
Let X be a proper smooth connected curve over C. As in 2, we set
S,
where
monodromy.
where
x
X
S
-
is the set of singular points of
A
n
X
with nontrivial
Consider a second order linear differential equation
K(X)
is
A2f d2f + PI (x)df
x + Po(x)f 0
dx
nonconstant and
P0’ P1 (K(X). If I’ 2
are linearly
0
MIN HO LEE
664
A2f
independent solutions of
sm(A2)f
O, let
0
(m+l)-st order differ-
be the
ential equation whose linearly independent solutions are
m
m-1
m
m-i (o
e
I, e
2
e
ie2
2-
sm(A2)f
Then these solutions form a basis of the solution space of the equation
0
and the monodromy representation for this equation is
.
,.m
pA2
’1 ’2
where
GL(2,C)
PA2’eI’2: l(Xo’Xo)
A2
is the monodromy representation for
zm:
and
GL(2,C) ---GL(m,C)
m-th
is the
symmetric power homomorphism.
We shall assume that the universal cover of
H
plane
F\H
X0
and that
F
with
X
0
is the
Poincar upper half
SL(2,Z).
[l(Xo,X O)
We shall also assume
that
SL(2,R).
PA2’I’2(F)
:H---X
Let
If
f
be the projection map and choose
0
can pull the functions
el(z), c02(z)
denote by
H
functions on
e
l,
e
f
and
2
f (z)
2(z)
x
i+m+l. Z
Wsm(A2
(-i)
tx
m+l.i= J
m
(z O)
such that
Z
with
x
O.
K(X), then we
Z
to functions near
back via
z
0
which we
We set
via meromorphic continuations.
F(z)
H
0
We can consider these functions as meromorphic
f(z).
and
z
sm(A2)f
is the general solution of the equation
0
(dx)m+Idz + C.)l e(z)m+l-i
(see [8, p.32]).
(2,m)
Now we shall determine a mixed automorphic form of type
solution
f
sm(A2)f
Z,
Z
LE(Sm(A2)).
As an analogy with Eichler [I] and Stiller [8, p.32], we set
.I
PROPOSITION 3. I.
F(z)
for some constants
PROOF.
g(z)
We have
z
+ m+l
ai=
g(t)(e(z)-e(t))mdt
Zo
c.
1
cie(z)
d
_dm_F
z dem ).
m+l-i
C.
By integration by parts we have
;z
z
d
(
0
dmF
(t) )(00(z)-e(t))mdt
-(e(z)-e(ZO))
de(t)
+
(e(z)-(t))
m
z
Applying integration by parts
becomes
associated to a
to the differential equation
m-1
0
m-I
m
dmF
m
d
(z O)
dmF(t)
de(t)
m
de(t).
times, the right hand side of the above equation
665
MIXED AUTOMORPHIC FORMS AND DIFFERENTIAL EQUATIONS
dm-IF
dmF
-((zl-(Zo ))m dm (Zo)
+ (- I)
+
(-1)2m(,(z)-(Zo)) m-1 dm-I
+
(Zo)
3m (m- I) ((z)-(Zo)) m-2
dF
(-l)mm(m-l)
2(co(z)-(Zo))(z O)
+ m!
I
z
z
dF(t)
d(t)-
d (t’
0
Hence we have
iz
.
for
z
C.
i
g(t)(o(z)-co(t))
F,
oo(z)m+l-i + m!F(z)
m+l
i=l i
d(t)
0
Thus the proposition follows.
PROPOSITION 3.2.
to
m
and
PROOF.
g(z)
is a mixed automorphic form of type
(2,m)
associated
0A2,coi,co2.
dm+IF(z)/d
Since
(z)
m+l
is a generalized automorphic form of weight
m+2 (see [8]), we have
d
d(yz)
dmF(yz)
dm+IF(yz)
dco(yz )m
dco(Tz)
d(yz).
d(yz)
m+l
(Cl(Z)+dl )m+2
dm+
F (z)
doo(z)
(cz+d)2(Cl(Z)+dl)
m
m+l
d(
dz
(Cl(z)+dl)
-2
2 d(z)
dz
(cz+d)
dmF (z)
d(z)
m
),
where
y
g(z)
Thus
r
E
is a mixed automorphic form of type
F,
DEFINITION 3.3.
(2,m)
if
Z
determined by
Sm(A2)Z
If
sm(A2)f
for some
Z’
Z
(2,m)
associated to
0A2’l’m2"
and
LE(Sm(A 2)),
Z
SL(2,Z).
0A2,I,2()
and
the mixed automorphic form
is said to be of the second kind.
K(X), then
g(z)
g(z)
of type
Furthermore,
is said to be exact.
Thus the cohomology group
HLI
E
(r
Vsm (A 2) ,xO)
can be described as the mixed automorphlc forms of type
(2,m)
of the second kind
modulo exact forms.
4.
PERIODS OF MIXED AUTOMORPHIC FORMS.
In the previous section we associated a mixed automorphic form
(2,m)
to each differential equation of the form
In this section we shall express the cocycles in
sm(A2)f
Z,
Z
E
g(z)
of type
LE(Sm(A2)).
MIN HO LEE
666
HLE (F, VSm (A2) ’x O)
corresponding to mixed automorphic forms of the second kind in terms of the periods
of mixed automorphic forms,
Let
f
be a solution to the differential equation
sm(A2)f
If the analytic continuation of
f
around
m
f +
ay
then the vector
(a
alm
F
]
is
m
+
+
Zl(Xo,Xo)
am+l 2,
is a cocycle in
am+l
I’
LE(Sm(A2)).
Z
Z,
HLE(F’Vsm(A2) ,x O)
(see 2).
P
Let
m
(uz- vw)
If
ym:
(m+1)x(m+1)
be the
(um,um-I v
m
SL(m+I,C)
SL(2,C)
matrix defined by
vm)
m-1
Pm t(wm’w
m
z ).
z
m-th symmetric power homomorphism, then we
is the
hav e
tzm(
SL(2,C).
o
for each
ay
If
p
m
y,m()
p
P
M
m
p-1
m
m
m-th symmetric power of the
is the
(_l)mpm
m
HLE(F,Vsm(A2),Xo
is a cocycle in
a y =m! M
where
tp
m’
t
we set
ay
M
(2x2)-matrix
(see [8, p.46]).
Then we have
a P + Mm a P
aP
aP
hence
F;
y,
for
is a cocycle in the parabolic cohomology group
([i], [6]).
PROPOSITION 4. I. The assignment
Eichler and Shimura
-
Hp(F Vsm{A 2) ’x0)
P}Y
E(r,Vsm(A2),Xo Hp (r,Vsm(^2),Xo)
PROOF.
{ay}
{a
determines an isomorphism
See
PROPOSITION 4.2.
P
ay
We have
t
P
P
(a.y ,1,...,ay,m+1)
_m
My )m! P-m
+ (I
t
C,
where
a
and
C
(el
PROOF.
Cm+l
P
,i
em+l
(-I)mm!
m
(m-i+l)
%+2-i
I Wsm(A
Y
2
x
Z
dx
,x
is a constant row vector.
See [8, Proposition 3 bis. 13 and 14].
PROPOSITION 4.3.
Wsm (A2) ,x
PROOF.
1’2’ ..m!
See [8, Theorem 3 his. 5].
of
A
m+l)/2
x
i
m+l
667
MIXED AUTOMORPHIC FORMS AND DIFFERENTIAL EQUATIONS
PROPOSITION 4.4.
PROOF.
(m-I))
I)2
i,x
[m
wm m-l)/2 001i-I z-i+lm
i-I
See [8, Theorem 3 bis. 15].
g(z)
Let
PROPOSITION 4.5.
be the mixed automorphic form of type
(2,m)
associated to the differential equation
sm(A2)f
LE(sm(A2)).
Z
Z,
Then we have
(-1)
g(z)
m+2
2(z)mz(z) (dx)
m
dz
WA2 ,x (z)
PROOF.
We have
dmF
z
m!(
m
(-I
z
d
0
Z
)m+2 l,x (dx)
z
Wsm A2) ,x
dz
+
Cl
so that
dmF
dm
d
z
g(z)
(-I
)m+2m,
I ,xZ
.dx)
Wsm (A 2) ,x
By Proposition 4.3 and Proposition 4.4, we have
Wsm (A2) ,x (z)
..m’(WA2 ,x (z))
I’2’
m(m+l)/2
I ,x l!2!’’’(m-l)!wmm-l)/2"
A sx
Thus it follows that
g(z)
(-i)
2(z)mz(z) (dx)
dz
m+2
WA2 x (z)
THEOREM 4.6.
Let
be the mixed automorphic form of type
g(z)
(2,m)
to the differential equation
Sm(A2)f
Z,
Then the associated parabolic cocycle
P
ay
for
i
P
(ay ,I’
{a P}
Y
P
ay ,m+l
Z e
LE(Sm(A2)).
is determined by the formula
+ (I
m
,.y)m.
p-Ira
t
c
l,...,m+l.
PROOF.
By Proposition 4.2, it suffices to show that
m
(m-i+l)
Y
Wsm (A 2) ,x
Using Proposition 4.2 and Proposition 4.4, we have
g(z)m(z)
Idz.
associated
668
MIN HO LEE
i
Y
---xZ
dx
Wsm (A 2) ,x
(-l)--m (ira_)
m
I
m
(_l)m+2 2
Z
i-I
wm+A2,x
Y
dx.
By Proposition 4.5, we have
m
(-I)
m+2
2
Z
wm+l
A2,x
g(z)dz
dx
from which the theorem follows.
This result should be compared to
[8, Theorem 3
bis. 17].
REFERENCES
I.
Eichler, M. Eine Verallgemeinerung der Abelschen Integrale, Math. Z. 67 (1957),
267-298.
2.
Hunt, B. and Meyer, W. Mixed automorphic forms and invariants of elliptic
surfaces, Math. Ann. 271 (1985), 53-80.
3.
4.
Lee, M. H. Mixed automorphic forms and holomorphic forms on elliptic varieties,
Pac. J. Math. 132 (1988), 363-370.
Lee, M. H. Mixed cusp forms and Hodge structures, to appear.
5.
Lee, M. H.
6.
Shimura, G.
Periods of mixed cusp forms, to
Sur les
intgrales attaches
appear.
aux formes automorphes, J. Mmth. Soc.
Japan II (1959), 291-311.
7.
{okurov,
8.
Stiller, P.
V. Holomorphic differential forms of higher degree on
varieties, Math. USSR Sb. 30 (1976), 119-142.
Kuga’s
modular
Special values of Dirichlet series, monodromy, and the periods of
automorphic forms, Mem. Amer. Math. Soc. No.299 (1984).