spherical polynomials and the periods of a certain modular form

Spherical Polynomials and the Periods of a Certain Modular FormAuthor(s): David KramerSource: Transactions of the American Mathematical Society, Vol. 294, No. 2 (Apr., 1986), pp.595-605Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/2000202 .

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Velume 294, Number 2* Aprll 1986

SPHERICAL POLYNOMIALS A1ND THE PERIODS OF A CERTAIN MODULAR FOIlM

BY

DAVID KRAMER

ABSTRACT. The space of cusp forms on SL2(Z) of weight 2k is spanned by certain modular forms with rational periods.

Kohnen and Zagier [5] have investigated those modular forms on SL2(Z) whose periods (in the sense of Eichler-Shimura) are rational. As an example they consider the function

fk,D(Z) = C E 2 1 k (s,bsc) (az + bz + c)

where D is a positive integer, k an integer greater than one, C a constant, and the summation over all triples of integers (a, b, c) with b2 _ 4ac = D. This function was introduced by Zagier in [13], where it was shown to be a cusp form of weight 2k on SL2(Z), and was later shown by Kohnen [4] to be the Dth Fourier coefficient of the holomorphic kernel function for the Shimura-Shintani correspondence between modular forms of half-integral and integral weight. In [6] it is shown that the fk D

span S2ks the space of cusp forms on SL2(Z) of weight 2k, if and only if L(f, k), the L-series associated to the modular form f, does not vanish at s = k, the center of the critical strip, for every Hecke eigenform fv It is at present only a conjecture, supported by some numerical evidence, that the fk D span S2k for all even k (for k odd, fk D = °). We study the related functions fk D A(Z) where now the summation is restricted to quadratic forms (a, b, c) belonging to the SL2(Z) equivalence class A. We prove that certain finite sets of the fk D A span S2k The method of proof relies on the special structure of certain ideal classes and does not indicate how the conjecture that the fk D span S2k mlght be settled.

A similar result for modular forms on an arbitrary discrete cocompact subgroup of SL2(R) has been obtained by Svetlana Katok [3].

Let be the set of all narrow equivalence classes of primitive binary quadratic

, > . . .. . . . . . .

torms ot posltlve olscrlmlnant. Throughout k wlll denote a posltlve lnteger, and we set w = 2k- 2. Let A be an SL2(Z) equivalence class of quadratic forms. We define the following classes associated to A:

aA = {(-c, -b -a) [(a, b,c) E A}, A' = {(a, -b,c) [(a, b,c) E A}.

Received by the editors May 1,1985. 1980 Mathematics Subject Classification (1985 Revision). Primary llFll, llF12; Secondary llC08.

r)1986 American Mathematical Society

0002-9947/86 $1.00 + $.25 per page

595



596 DAVID KRAMER

Let Pk denote the C-vector space of homogeneous polynomials in two variables of degree w. For an equivalence class A E a? of forms of discriminant D, let

Pk,D,A(X y) = E Q(x yk-1 QeA

with the summation over the reduced forms of A (a binary quadratic form Q(x, y) = ax2 + bxy + cy2 is reduced if a > O, c > O, b > a + c), and then set

Pk, D, A Pk, D, A _ Pk, D, SA

We shall study the span in Pk of Pk+D AS A E X, and prove a theorem about the space of modular forms on SL2(Z) of weight 2k.

DEFINITION 1. A polynomial mr(a, b, c) is called spherical with respect to b2 _ 4ac if it satisfies

( db2 - aaaC ) 7T(aS b, c) = O.

PROPOSITION 1. Let dn k(a, b, c) be defined by

2k-2

L d ( b ) tI 2k-2-n ( 2 + b + 2)k-1 sl =O

The set { d n k(a, b, c), O < n < w } is a basis for the space of homogeneous spherical polynomials of degree k-1.

For a proof see [8, the corollary to Theorem 18].

THEOREM 1. Let sk+ be the subspaces of Pk spanned by the collections { Pk+D A +

Pk,D,A'}, A E S? I. Then the dimensions °f Sk+ are equal to T)k-+, where

+_lk/2+1, keven, rlk - \ (k + 1)/2, k odd,

and

f k/2, keven,

\(k+1)/2, kodd.

II. There is an infinite subset of the set of equivalence classes of forms, which we

shall construct explicitly, that has the following property: any 2k classes in this set have corresponding Pk D X + Pk+D At which span Xrk+. Moreover, given any positive discriminant J9, we can choose these classes such that each has discriminant f 2D for some integer f; i. e., all of the equivalance classes are associated to module classes in the same real quadratic field.

NOTE. We consider Pk+D A + Pk+D A! rather than Pk+D A alone, because in the course of the proof we need to assume that our summation includes the form



597 SPHERICAL POLYNOMIALS AND A CERTAIN MODULAR FORM

(c, b, a) whenever it contains (a, b, c). Let

£A- mean E + L + L + , A A' SA SA'

where the summation is over the reduced forms in the given class. Since the coefficients of Q(x, y)k-l are the dn k(a, b, c), it follows from Proposition 1 that to establish the dimension of jYk+ it suffices to find w + 1 - T1k- Q-linearly independent linear combinations of the dn kla, b, c), for each of which £,A- = O for all A E X, and T1k- Q-linearly independent linear combinations of the dn k(a, b, c), of which no nontrivial linear combination sums (x£,A-) to zero for all cycles A E a?.

We first establish the requisite number of relations. Let

(l) <)n,k(a b, c) = dn,k(a b, c) - dw-n k(a, b, c), O < n < w, and

(2) An-k(a,b,c) = dn k(a,b - 2a,a - b + c) - dn k(c2c - b,a - b + c)

+(-1)k [dn k(a - b + c, b - 2a, a ) - dn k(C - b + a,2c - b, c)],

O < n < w. We assert that for any A E a?

(i) ° = SA fn k(a, b, c), (ii) O = LA vJn-k(a b, c).

That (i) is true follows immediately from the facts that dw-n k(a, b, c) = dn k(c b, a)

and (a, b, c) E A (c, b, a) E A'. To establish (ii), let Rn4 k(a, b, c) = dn k(a, b, c)

+ (-l)kdn k(C b, a). Then Rn-k(a, b, c) = +Rn-+k(-c -b, -a), and the result follows from [7, Proposition 3']. From among the polynomials (1) and (2) we select W + 1 - T1k- that are independent.

LEMMA 1. The collections

k = {+n,k° < n < k - 1; An+k1 < n < k - 1, n odd } and

k = { fn,k s 0 < n < k - 1 ; An,k s 1 < n < k - 1 , n odd } are each composed of linearly independent polynomials.

PROOF. From Proposition 1 it is clear that the fn k ° < n < k-1, are linearly independent. Moreover, fn k(a b, c) = -+n k(C b, a), while for n odd, An-k(a, b, c) = ++n-k(c,b,a). Thus it suffices now to show that the An-k(a,b,c) in Sk+ are independent among themselves.

To establish this we introduce a change of basis for the space of spherical polynomials, namely the transformation

(a, b, c) (A, B + 2A, A + B + C) .



598 DAVID KRAMER

Then

++k(a,b,c) = ++k(A,B + 2A,A + B + C)

= dn k(A, B, C)-dn k(A + B + C, B + 2C, C)

+ (-l ) kdn k ( A , B, C)-dn k( C, B + 2C, A + B + C )

= dn k(A B, C)- E ( n _J )dJ k(A, B, C)

+ (-1 ) ( dw - n ,k ( X 7 Be C ) _ E ( w j ) dy k ( A, B, C ) ),

which for 1 < n < k - 1, n odd, is a linear combination of the dJ k(A, B, C) with O < j < w-n-1 and the coefficient of dW_"_l J not equal to zero (except for Ak+- l kS k even, which is zero). This triangularization of the { +k in k- demon- strates their linear independence.

The number Of fn k in k- iS k-1 and the number of + + in Sk- iS given in Table 1.

TABLE 1. The number of 14n,k in k

S?+ ,_

k even k/2 - 1 k/2 kodd (k-1)/2 (k-1)/2

Thus in all cases we have, as asserted, w + 1 - T1k independent relations. We complete the proof of Theorqm 1 by proving the following: Consider the

family of discriminants Dt = t2 + 4t, t = 2, 3, 4,.... Belonging to Dt is a wide equivalence class of quadratic forms associated to the reduced (in the wide sense) number

[[lSt]]=l+t+ 1

1 + * * -

(For a discussion of the relationship between continued fractions and reduced quadratic forms see [12].) Since this "+"-continued fraction is of even length, the wide equivalence class consists of two distinct narrow equivalence classes, At and SAt, whose cycles of reduced forms are generated by the "-"-continued fractions ,uO = ((3, 2, 2, . . ., 2)) and 0 = ((t + 2)), respectively.

PROPOSITION 2. Ser

(3) gn k(t) = 2 E d",k(a b - 2a, a - b + c). A,

Then any nontrivial linear combination of elements of the set (of cardinality T)k-)

(gn-k(t), n odd, 1 < n < k - 1 (< k - 1 if (-l)k = _ 1); gW+k(t))

is zero for at most 2k vaolues of t > 2.




PROOF. The cycle of reduced forms of At is { Qp = (ap, bp, cp)}O < p < t- 1 where

ap = ( p + l) t _ p2, bp = (2p + 3)t - 2( p2 + p ), cp = ( p + 2)t -- ( p + 1)2.

SAt contains the single reduced form Q = (lSt + 2,1). (Note that At - At, so SAt = 2(LAt _ £A,).) Taking n odd, and recalling that

dn k(a? b, c) = (-l)ndn k(a -b, c), we have

t-1

(4) Ef dn k(a, b - 2a, a - b + c) - fi dn k((P + l)t - p2, t-2p, -1). A, p=O

Under the transformation p t - p the right-hand side of (4) becomes

t t

E dn k((P + l)t -p ,2p - t,-1) - - S dn k((P + l)t -p St - 2pS-1) p=l p=l

t-1

- - 5£. dn k((P + l)t _ p2, t - 2p,-1) + dn k(t, t,-1) -dn k(t,-t,-1), p =O

from which we conclude that

E dn k(a, b-2a, a-b + c)-2(dn k(t, t, -1)-dn *(t, -t, -1)) At

- dn k(t, t,-1).

We also have

S dn,k(a, b-2a, a - b + c) = dn,k (1, t, -t ) . sAt

Thus l:or n odd,

(, + )dn k(a,b- 2a,a - b + c) = dn k(tvtS-1) + dn k(l,t,-t) = gn+k(t) A SA

Writing

d ( b ) ^ (k-1 )(k-1-8) ab k-(l+a+#@)

a + ,BA k + 1

we have

(5) + ( ) , ( k - 1 )( k - 1 - IB )( 1) k (l +a+A)( a+/? k 1-a

(X +,(3<k -1

If n = k-1, then (x + ,8 - k - 1 - xx, and so g"+k(t) - 0. Otherwise, for n < k - 1, n odd, we claim that the right-hand side of (5) is a polynomial in t of degree < k-1 whose tln+l/2 coefficient is nonzero but all of whose lower powers of t

vanish. To see this, observe that xx + ,B = n - a, and so cY + ,8 and k-1 - a are each

minimal for the largest possible choice of a, namely for,8-1, a-(n - 1)/2. Then a + /3 - (n + 1)/2, and k - 1-a = k - ((n + 1)/2)> (n + 1)/2, since n < k - 1.



600 DAVID KRAMER

We now consider t-l

gw k(t) = E dW k((P + 1)t -p, t- 2p,-1) + dW k(l, t,-t) p =o

t-l

= (+1) + , (( p + 1)t -p2)k-1 p=O

We claim that the sum in this expression is a polynomial in t with no constant term, whence gw+k(t) does have a constant term, namely + 1.

t-1 t-l , (( p + 1)t _ p2)k-1 = L (+pW + higherpowers of t).

p=o p=o

But

£ pw = BW+l(t)-Bw+l p=o w+ l '

where Bs(t) is the sth Bernoulli polynomial. Since w + 1 is odd, this is a polynomial in t with no constant term, and our claim has been proved.

This triangularization of the polynomials in (3) shows them to be linearly independent.

The gn+k(t) for n odd have degree at most k - 1, while the degree of g+k(t) is 2k-1. Thus any nontrivial linear combination of the gn+k is a polynomial in t of degree at rsost w + 1 = 2k - 1 and so can be zero for at most 2k - 1 values of t. Thus for any 2k values of t > 2, the corresponding set of the Pk+D A must span k

To show that we can choose the At to be associated to the same real quadratic field, fix a discriminant D > O. The class At has discriminant t2 + 4t, and since t2 + 4t = f 2D is equivalent to (t + 2)2 _ f 2D = 4, which is just Pell's equation, we can find infinitely many values of t for which At has discriminant f 2D (for different values of f ).

We note the following corollary, which we shall need in the proof of a later theorem.

COROLLARY. The sets { ++ k(a, b, c)}, n odd, 1 < n < k-1, and { 74n k(aS b, c)}, n odd? 1 < n < k - 1, are bases for the spaces of those spherical polynomials P of degree k-1 for which £X-+ P = O for all A E 52.

We now consider the function (_l )(k-1)/223k-2D k-1/2 1

fk,D(z) - tZ7(2k-2k-1) (,b,c) (aZ2 + bz + c) where D is a positive integer, k an integer > 2, and where the summation is over all triples of integers (a, b, c) with b2 _ 4ac = D. This function was introduced by Zagier in [13], where it was shown to be a cusp form of weight 2k on SL2(Z). Then Kohnen showed in [4] that fk D iS the Dth Fourier coefficient of the holomorphic kernel function for the Shimura-Shintani correspondence between modular forms of weight k + 2 and weight 2k. We shall be interested in the functions fk D A(Z)




defined in the same way as fk D, except that here the summation is over only those forms (a, b, c) belonging to the equivalence class A. The periods of the functions fk,D,A = fk,D,A - fk,D,A' were studied by Kohnen and Zagier in [5]. Let us here recall the basic facts about periods of modular forms.

Let f be a cusp form of weight 2k. Then for any integer O < n < 2k - 2 = w, we can define the nth period of f,

rn(t) = I f(z)z dz o

The fundamental theorems on periods are the following, our exposition being essentially that in Lang [9].

THEOREM 2 (EICHLER- SHIMURA). There are certain relations among the periods of modular forms:

(i) rS +(-1)5 (y)rw-s+ +(-1)5 5£ ( J )rJ = O;

y even I-s mod 2

(ii) L (y)rw-s+v + L ( J S)r-O y odd jWs mod 2

Since rJ(t) is real if j is odd, and pure imaginary if j is even, these relations can be separated into two sets of relations: one involving only the even periods, the other only the odd. These relations then define two subspaces, V+ of 9)k+ and V- of 9)k where 9iJk+ (resp. 9iJk) is the subspace of 9)k consisting of polynomials even (resp. odd) in both variables.

THEOREM 3 (EICHLER- SHIMURA). Let S2k be the space of cusp forms of weight 2k on SL2(Z). Define the mappings r - of S2k into 9)k- by

(6) r-( f )(x, Y) = t+ E ( n )rn( t )x y OAnAw

(_l)n = + 1

where e+= -i, e_= 1. Then r- is an isomorphism of S2k and V-, while r+ is an isomorphism of S2k with a

subspace of V+ of codimension 1, not containing xw _ yw.

Recall that Pk D A(X Y) = EA Q(x y) In [S] the following theorem is proved.

THEOREM 4. Let k > 2, and let A be an SL2(Z) class of quadratic forms of discriminant D > O, D not a square. Then

(_l )(k + 1)/2

23k + 1 k, D,A

Pk,D,A(X + y, X) Pk,D,A(X Y, Y) + (-1) Pk D SA(X,-X + y)

Pk D sA( Y, x + Y) B ) (A 1 - k)(xW _ yW)



602 DAVID KRAMER

or, equivalently, for (-l)n = + 1,

2 3 k-1 ( n ) n ( t k, D, A )

= E dnk(c-b+a,2a-b,a)-dnk(a,b-2a,a-b+c)

( cl , 1v , c) E A reduced

+(_l)k (dn k(a-b + c,b-2c,c)-dn k(co2c-b,c-b + a))

( cl , 1v, c) E SA reduced

+ ( t;w,n-t;O,n ) B g( A, 1-k ),

where Bs is the sth Bernoulli number, /8Jk the Kronecker delta, and t(A,s) the

Dedekind zeta function of the ideal class associated to A.

We can now state and prove our second result.

THEOREM 5. Let fk D A be defined as above. For k an integer greater than 1, the

collection of the fk D A, A E 5{, spans S2k, the space of cusp forms on SL2(Z) of

weight 2k.

PROOF. BY Theorems 3 and 4, if suffices to show that the polynomials r+( fk+ D A)

span the image of r+(S2k) as defined in Theorem 3.

LEMMA 2. fk-D,A = (-1) fk,D,AA-

PROOF. If (a, b, c) E A, then (-a-b,-c) E SA'. Thus fk D A = (-l)kfk D fRA', and

the result follows at once from the definition of fk-D A.

From Lemma 2 and Theorem 4, as well as the fact that (a, b, c) E A (c, b, a)

E A', we immediately deduce

LEMMA 3. Let

Sn k(a,b,c) = -dn,k(a - b + c,b - 2a,a)

-dn k(c-b + a, b-2c, c) + dn k(c b-2c, c-b + a).

The even periods °+fk+ D A are given for 2 < n < w-2 by

-(-1 (k + 1)/2( W ) 23k-3 n k D,A

= (, +(-l)kE,)(-dn k(a - b + c,b - 2a,a) - dn k(c - b + a,b - 2c,c)

A SA

+dn k(c, b - 2c, c - b + a))

=: ( E + (-1) E )Sn k(a, b, c). A SA

Now, by the corollary to Theorem 1 and Lemma 3, there is a relation among these

even periods Of fk+ D A if and only if some linear combinations of the Sn k, n even,

equals a linear combination of the Sn k, n odd. Thus to prove Theorem 5, it suffices



SPHERICAL POLYNOMIALS AND A CERTAIN MODULAR FORM 603

to show that the dimension of the subspace of the sphencal polynomials of degree k-1 spanned by the Sn k has dimension equal to the span of the Sn k n odd (which is [(k - 1)/21) plus the dimension of S2k (which is [k/6], unless 2k - 2 (mod 12)7 in which case it is [k/6] - 1), plus 1 (for the zeroth period), which totals [2k/3] in all cases. We proceed now to calculate this dimension. For O < n < w define polynomials Kn k(X, y) in 9)k by

n,k( ( Y) (X y) (-X) w n +_ (-X) n( ) w-

-(X -y) yw-n -y"(X _ y)W-n

LEMMA 4 Sn=O Sn,k(a b, c) = En=O dn k(a, bS C)Kn k(XS y)

PROOF. w

, Sn k(a, b, c)xnyW-n n =O

w

= . (dn k(a,b - 2a, a - b + c) + dn k(c b - 2c,c - b + a) t1 = O

-d n k ( a-b + c, b-2a, a )-dn k ( c-b + a, b-2c, c))xnyW-n

= ((a-b + C)X2 +(b-2a)xy + ay2)k-

+ ((c - b + a)x2 + (b - 2c)xy + Cy2)

-(ax2 +(b - 2a)xy +(a - b + C)y2)k-

-(CX2 +(b - 2c)xy +(c - b + a)y2)k-

w

= E (dn,k(a,b,C))((X_y)n(_X)W-n+(_x)n( _ )W-

n =O

-(X -y)nyw-n -y"(X _ y)W-n

w

= E dn k(a beC)Kn,k(X' Y)-

n =O

We now assert that the dimension of the span of the Rn k in the space of spherical polynomials of degree k- 1 is equal to the dimension of the subspace of 9)k

spanned by the «(n k(Xs Y)*

PROPOSITION 3. Let V and A be vector spaces over a field, of dimension N, with bases v1, . . ., Vy and al,. . ., aN respectively, and suppose that

SV N

, wi @ al = 5£ VJ 2) bJ,

1=1 J=1

for some wl, 1 < j < N, in Vand b>, 1 < j < N, in A. Then the dimensions of the subspace of V generated by the wi, and of the subspace of

A generated by the bj are equal.



604 DAVID KRAMER

PROOF. Write, for 1 < i < N, N N

wl= E aiJuJ and bl= E jBZJaJ. J=1 J=1

Then EN=lWi @ ai = EN=lVi @ bl implies Aij = AJI- Hence the matrices (IJ) and

t(iBzy) are equal, whence (OIIJ) and (/2IJ) have the same rank. Applying Proposition 3

to A, the space of spherical polynomials, V = gak an = Sn k(a b c) and vn =

Kn k(x, y) proves the assertion. For a matrix y = (a b) in SL2(Z) we define an endomorphism Il(y) on 90)k as

follows: for any polynomial P(x, Y) E 90)k let

(7)P(x, y) fn k((X Y)7),

where

(x, y)y = (x, Y)( c d) = (ax + cy, bx + dy).

Now let

t=(l 1) and ff (1 O)

Let t be the image Of 90)k under t = (1 - a)(t - t2). Then a simple calculation

shows that t is precisely the span of the Kn k. In calculating the dimension of @, it

makes no difference if we consider the action of SL2(Z) on bak(C) instead of on

90'k(Q) The operator Il(t) has two eigenspaces on the homogeneous polynomials of

degree 1: C(x + py) with eigenvalue -p, and C(x + p-y) with eigenvalue -p-

(p = e(2Xi/3)), while Il(a) takes (x + py) to p(x + p-y) and (x + p-y) to p-(x + py).

Let A = x + py, B = x + p-y. We have the direct sum decomposition k-1

9@w(C) = @ C(AJBW-J @ AW- J =O

into {-invariant subspaces, with

II(+)(AjBW-J + AW-JBJ) = pJp-w-j(l-pJp-W-J)(l-pJp-W-J)AJBW-J

+pw-jp-J(l-pJp-W-J)(l-pW-Jp-J)AW-JB8,

which is zero if and only if pJp-W-J = 1. But

pi p-w-J = p2W-J = l j - 2w (mod 3) -

Thus the dimension of t is

#{ jl°< jAk-1;2w-j--lor2(mod3)}=(2k/3),

as was to be shown.

REFERENCES

1. Erich Hecke, Zur Theorie der Elliptischen Modulfunktionen, Math. Ann. 97 (1926), 210-242.

Number 23 in Hecke, Mathematische Werke, Vandenhoeck & Ruprecht, Gottingen, 1959.

2. , Analytische Funktionen und Algebraische Zahlen, Zweiter Teil, Abh. Math. Sem. Univ.

Hamburg. 3 (1924), 13-236. Number 20 in Mathematische Werke.

3. Svetlana Katok, Modular forms associated to closed geodesics and arithmetic applications, Bull. Amer.

Math. Soc. (N.S.) 11 (1984), 177-179.



SPHERICAL POLYNOMIALS AND A CERTAIN MODULAR FORM 605

4. Winfried Kohnen, Beziehungen Zwischen Modulformen Halbganzen Cewichts und Modulformen Can7en Cewichts, Schriften Nr. 131, Bonner Math., Bonn, 1981.

5. Winfried Kohnen and Don Zagier, Modular forms with rational periods (to appear). 6. , Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981),

175-198. 7. David Kramer, Applications of Causs 's theory of binary quadratic forms to zeta functions and modular

forezs, Trans. Amer. Math. Soc. (to appear). 8. Andrew Ogg, Modular forms and Dirichlet series, Benjamin, New York, 1969. 9. Serge Lang, Introduction to modular forms, Springer-Verlag, Berlin-Heidelberg-New York, 1976.

1(). Carl L. Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Gottingen Math.-Phys. K1. II 1969, 87-102.

11. Don Zagier, A Kronecker limit formula for real quadratic fields, Math. Ann. 213 (1975), 153-184. 12. , Zetafunktionen und Quadratische Korper, Springer-Verlag, Berlin-Heidelberg-New York,

1981. 13. , Modular forms associated to real quadratic fields, Invent. Math. 30 (1975), 1-46.

DEPARTMENT OF MATHEMATICS, SMITH COLLEGE, NORTHAMPTON, MASSACHUSETTS 01063



spherical polynomials and the periods of a certain modular form

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