DOI 10.1007/s11139-011-9341-y
A basis for Sk
(Γ
0(
4)) and representations of integers
as sums of squares
Shinji Fukuhara· Yifan Yang
Received: 11 February 2011 / Accepted: 18 August 2011 / Published online: 11 February 2012 © Springer Science+Business Media, LLC 2012
Abstract In this paper, we find a basis for the space Sk(Γ0(4)) of cusp forms of even weight k for the congruence subgroup Γ0(4) in terms of Eisenstein series. As an application, we obtain formulas for r4s(n), the number of ways to represent a nonnegative integer n as a sum of 4s integral squares.
Keywords Modular forms (one variable)· Period polynomials · Fourier coefficients
of modular forms· Sum of squares
Mathematics Subject Classification (2000) Primary 11F11· Secondary 11F30 ·
11F67
1 Introduction and statements of results
Throughout the paper, we assume that k is an even positive integer. Let Γ be a con-gruence subgroup of SL2(Z), and let Mk(Γ ) and Sk(Γ ) be the space of modular forms and the space of cusp forms of weight k on Γ , respectively.
The first author was partially supported by Grant-in-Aid for Scientific Research (No. 22540096), Japan Society for the Promotion of Science. The second author was partially supported by Grant 99-2115-M-009-011-MY3 of the National Science Council, Taiwan (ROC). Part of the work was done while the second author visited the first author at Tsuda College. He would like to thank Tsuda College for the great hospitality.
S. Fukuhara
Department of Mathematics, Tsuda College, Tsuda-machi 2-1-1, Kodaira-shi, Tokyo 187-8577, Japan
e-mail:fukuhara@tsuda.ac.jp
Y. Yang (
)Department of Applied Mathematics, National Chiao Tung University and National Center for Theoretical Sciences, Hsinchu, Taiwan 300
For Sk(Γ0(2)), we have given in [4] a basis{E2ji∞Ek0−2j| j = 2, . . . , d + 1}, where Eni∞ and En0 are normalized Eisenstein series of weight n for the cusps i∞ and 0, respectively, and d= dim Sk(Γ0(2))= k/4 − 1. The existence of such a basis was suggested in [1,6]. In this paper, we will find a basis for Sk(Γ0(4)). The main motivation is to obtain formulas for rs(n), the number of ways to represent a non-negative integer n as a sum of s integral squares.
To state our main results, let us first recall that the group Γ0(4) has 3 cusps, rep-resented by i∞, 0, and 1/2. To each cusp α and each even integer k ≥ 4, we may associate an Eisenstein series
Eα k(τ ):= d/c∼α 1 (cτ− d)k,
where the sum runs over all cusps d/c equivalent to α under Γ0(4). More explicitly, we have Eki∞(τ )= 1 2 (c,d)=1,4|c 1 (cτ− d)k = 1 2k− 1 2kEk(4τ )− Ek(2τ ) , E0 k(τ )= 1 2 (c,d)=(c,4)=1 1 (cτ− d)k = 2k 2k− 1 Ek(τ )− Ek(2τ ), E1/2 k (τ )= 1 2 (c,d)=1,2|c,4c 1 (cτ− d)k = 1 2k− 1 −Ek(τ )+ 2k+ 1Ek(2τ )− 2kEk(4τ ) , (1.1) where Ek(τ )= 1 + (2π i)k Γ (k)ζ (k) ∞ n=1 σk−1(n)qn= 1 − 2k Bk ∞ n=1 σk−1(n)qn, q= e2π iτ,
is the Eisenstein series of weight k on SL2(Z) and Bk is the kth Bernoulli number; see Lemma 3.2 of [10] for calculation of Fourier expansions of the Eisenstein series. In fact, the series Sk(0, 1), Sk(1, 0), and Sk(1, 1) in Lemma 3.2 of [10] are essentially our Eki∞, Ek0, and E1/2k here, respectively. This is because Γ0(4) is conjugate to Γ (2) by2 00 1. These Eisenstein series have the property that
lim τ→i∞ Eα kkγ (τ )= 1 if a/c∼ α, 0 if a/c∼ α, for all γ =a bc d∈ SL2(Z). In particular, if α ∼ β, then Ekα
1E β
k2 is a cusp form of weight k1+ k2on Γ0(4).
In order to simplify the expressions in the statements of our theorems, we rescale the Eisenstein series and define
Eki∞(τ )= Eki∞(τ ), Ek0(τ )=Eki∞kW4
k 0
where W4denotes the Atkin–Lehner involution on Mk(Γ0(4)). In addition, for k= 2, we also define the Eisenstein series Eα2(τ )using the Fourier expansions given in (1.1). These Eisenstein series E2α(τ ) are not modular forms, but using the transformation property of E2(τ ), it can be easily verified that for any modular form f of weight k on Γ0(4), the functions E2i∞(τ )f (τ )− 1 π ikf (τ ), E0 2(τ )f (τ )− 1 π ikf (τ )
are modular forms of weight k+ 2 on Γ0(4). (See (3.6) and (3.7) below.) Now we can give our basis for Sk(Γ0(4)).
Theorem 1.1 Let k≥ 6 be an even integer. Then the sets
E2i∞Ek0−2− 1 π i(k− 2)E 0 k−2 ∪ Eni∞Ek0−n| n = 4, 6, . . . , k − 4 and E20Ek−2i∞ − 1 π i(k− 2)E i∞ k−2 ∪ En0Eki∞−n| n = 4, 6, . . . , k − 4
are both bases for Sk(Γ0(4)).
As mentioned earlier, our motivation to study Sk(Γ0(4)) is to obtain exact formulas for rs(n), the number of ways to represent a nonnegative integer n as a sum of s integral squares. (See [2,5] for surveys of the long and rich history of this problem.) To see the connection between Sk(Γ0(4)) and rs(n), let us recall that the generating function for rs(n)is Θ(τ )s= n∈Z qn2 s , q= e2π iτ.
When s is even, we have
Θ(γ τ )s=
−1
d
s/2
(cτ+ d)s/2Θ(τ )s
for all γ =a bc d∈ Γ0(4), where (−1d )is the Legendre symbol. Thus, for a positive integer s, the function Θ(τ )4s is a linear combination of Eisenstein series E2si∞(τ ), E2s0(τ ), E2s1/2(τ ), and the functions in Theorem1.1. In fact, we can do a little better.
The theta function Θ(τ ) satisfies
Θ − 1 4τ = 2τ i Θ(τ ). It follows that Θ4s|2sW4= (−1)sΘ4s.
In other words, Θ4s∈ M2s(Γ0(4), (−1)s), the (−1)s-Atkin–Lehner eigensubspace of M2s(Γ0(4)). Moreover, Θ(τ ) vanishes at the cusp 1/2. (This is because Θ(τ ) has the infinite product representation η(2τ )5/η(τ )2η(4τ )2. Thus, the zeros of Θ(τ ) must be at cusps. From the above transformation, we conclude that Θ(τ ) must vanish at 1/2.) Therefore, we have Θ(τ )4s∈ CE2si∞(τ )+ (−1)sE2s0(τ )⊕ S2s Γ0(4), (−1)s . Now we have dim SkΓ0(4),+ = k 4 − 1, dim SkΓ0(4),− =k 2− k 4 − 1.
From the dimension formulas and Theorem 1.1, we easily obtain bases for Sk(Γ0(4),±1).
Corollary 1.2 If k≥ 8 is an even integer, then
Eni∞Ek0−n+ En0Eki∞−n| n = 4, 6, . . . , 2k/4
is a basis for Sk(Γ0(4),+). In particular, if k ≡ 0 mod 4, then Θ(τ)2k is a linear
combination of Eki∞(τ )+ Ek0(τ )and the functions above.
Corollary 1.3 If k≥ 6 is an even integer, then
E2i∞Ek0−2− E20Eki∞−2− 1 π i(k− 2) E0k−2− Eik∞−2 ∪ Ein∞E0k−n− En0Eki∞−n| n = 4, 6, . . . , k − 2k/4 − 2
is a basis for Sk(Γ0(4),−). In particular, if k ≡ 2 mod 4, then Θ(τ)2k is a linear
combination of Eki∞(τ )− Ek0(τ )and the functions above.
We remark that since Γ0+(4) is conjugate to Γ0(2) by1 1/20 1 , we can first obtain a basis for Sk(Γ0(2)) and apply τ→ τ +1/2 to the basis to get a basis for Sk(Γ0(4),+), and consequently exact formulas for rs(n). This is the approach adopted in [6]. How-ever, this method only work for the cases 8|s. Also, the basis for Sk(Γ0(4),+) ob-tained in this way is different from the basis in Corollary1.2.
Another result of similar nature is given by K. Kilger. In his Ph.D. thesis [7], K. Kilger obtained bases for Sk(Γ0(N )), N = 1, . . . , 4, using modular symbols. His bases in the case N= 4 are similar, but different from ours.
Example 1.4 Here we give some formulas for r4s(n). In the following, we let fs,0:= Ei2s∞+ (−1)sE2s0, fs,2:= Ei2∞E 0 2s−2+ (−1)sE 0 2E2si∞−2− 1 π i(2s− 2) E2s0−2+ (−1)sE2si∞−2,
k 0 fs,n:= Ein∞E 0 2s−n+ (−1) sE0 nE i∞ 2s−n (n≥ 4, n even). By comparing suitably many Fourier coefficients, we find
Θ8= f2,0, Θ12= f3,0+ f3,2, Θ16= f4,0+ 17 16f4,4, Θ20= f5,0+ 17 31f5,2− 134 93 f5,4, Θ24= f6,0+ 43928 18657f6,4− 6848 18657f6,6, Θ28= f7,0+ 2073 5461f7,2− 1561873 737235f7,4+ 460309 245745f7,6, Θ32= f8,0+ 11379631232 4392213525f8,4− 13142016 6506983 f8,6+ 967923424 627459075f8,8, Θ36= f9,0+ 929569 3202291f9,2− 2997123429668 1165073523075f9,4+ 817033178804 317747324475f9,6 −130045826398 35305258275 f9,8.
We now indicate how Theorem1.1is proved. We shall see that Theorem1.1is, in fact, a consequence of linear independence of certain period polynomials of cusp forms on Sk(Γ0(4)).
For convenience, let us set w= k − 2. Assume that N is an integer with N > 1. For a cusp form f∈ Sw+2(Γ0(N ))and an integer n with 0≤ n ≤ w, we let
rn(f ):=
i∞ 0
f (z)zndz (1.3)
be the nth period of f . Since rn: Sw+2(Γ0(N ))→ C is a linear functional, there exists a unique cusp form RΓ0(N ),w,n(z)∈ Sw+2(Γ0(N ))such that
rn(f )= cw(f, RΓ0(N ),w,n), cw:= 2−1(2i)
w+1 (1.4)
for all cusp forms f of the same weight on Γ0(N ). Here
(f, g):=
Γ0(N )\H
f (z)g(z)ywdx dy, z= x + iy, (1.5)
denotes the Petersson inner product of f and g. We now explain the relation between RΓ0(4),w,nand Eni∞Ek0−n.
Using Rankin’s method [12] and following the argument in the proof of Proposi-tion 2 of [6], we can show that if f is a newform of weight k on Γ0(4), then for even
integers n > k/2, we have
f, Ein∞E0k−n= ck,nL(f, k− 1)L(f, n),
where L(f, s) denotes the L-function associated to f and ck,nis a constant depend-ing on k and n. (See Proposition3.1 below.) For oldforms from Sk(SL2(Z)) and Sk(Γ0(2)), there are also similar formulas. On the other hand, from the definitions (1.3) and (1.4) of rnand RΓ0(4),w,n, it is easy to see that
(f, RΓ0(4),w,n)= ck,nL(f, n+ 1)
for some constant ck,nindependent of f . Therefore, even though Eni∞Ek0−n is not precisely a multiple of RΓ0(4),w,n−1, we can still deduce linear independence among Eni∞E0k−nfrom that among RΓ0(4),w,n.
To obtain linear independence among RΓ0(4),w,n, we consider period polynomials r(f )which for cusp forms f ∈ Sk(Γ0(N ))for general N are defined by
r(f )(X):=
i∞ 0
f (z)(X− z)wdz.
Furthermore, even and odd period polynomials r+(f )and r−(f )are defined by
r±(f )(X):=1 2
r(f )(X)± r(f )(−X).
The period polynomials for RΓ0(N ),w,n are computed in [4] and will be crucial in our proof of Theorem1.1. To state the formula, we let Bm(x)(resp., Bm) denote the mth Bernoulli polynomial (resp., number). By Bm0(x), we denote the mth Bernoulli polynomial without its B1-term (see [9, page 208]):
Bm0(x):= 0≤i≤m i=1 m i Bixm−i= 0≤i≤m ieven m i Bixm−i.
For an integer n with 0 < n < w, let
˜n = w − n
and define a polynomial SN ,w,nin X by
SN ,w,n(X):=N ˜nXw ˜n + 1 B0˜n+1 1 N X − 1 n+ 1B 0 n+1(X).
Then the period polynomials r±(RΓ0(N ),w,n)are given as follows [4].
Theorem 1.5 [4, Theorem 1.1] Let N be an integer greater than 1. For an even
integer n with 0 < n < w, we have
k 0
Also, for an odd integer n with 0 < n < w, we have
r+(RΓ0(N ),w,n)(X) = SN ,w,n(X) − (w+ 2)Bn+1B˜n+1 (n+ 1)(˜n + 1)Bw+2 Xw N p|N 1− p−(n+1) 1− p−(w+2)− 1 Nn+1 p|N 1− p−(˜n+1) 1− p−(w+2) ,
where p runs over all prime divisors of N .
In the sequel, we focus on the case N= 4. Furthermore, we consider only vector spaces overC, and linear independence means that of over C. First, we will prove the following theorem:
Theorem 1.6 The polynomials
S4,w,n(X) (n= 2, 4, . . . , w − 2)
are linearly independent.
Note that an analogous result for Γ0(2) was obtained in [4], where explicit eval-uation of Hankel determinants formed by Bernoulli numbers is the main ingredient. Here the key to our proof of Theorem1.6is the 2-adic ordinal of the coefficients of S4,w,n(X). The method used here is not applicable to the case Γ0(2). (This is due to the fact that ord2(4)= 2, but ord2(2)= 1.)
By the similar argument as for proving Theorem1.6, we can derive the following result.
Theorem 1.7
(1)
{RΓ0(4),w,n| n = 1, 3, . . . , w − 3}
form a basis for Sw+2(Γ0(4)). (2)
{RΓ0(4),w,n| n = 2, 4, . . . , w − 2}
form a basis for Sw+2(Γ0(4)).
Remark 1.8 We now recall the formula
RΓ0(N ),w,n|w+2WN= (−1)n+1Nw/2−nRΓ0(N ),w,˜n
in [4, page 330] for the Atkin–Lehner involution WN. In Theorem1.7(1), the basis can be replaced by
deleting n= 1 and adding n = w − 1. These correspond to each other by the Atkin– Lehner involution.
Now, by Theorem 1.7, we know that f = 0 if (f, RΓ0(4),w,n)= 0 for all n = 1, 3, . . . , w− 3 (or n = 2, 4, . . . , w − 2, respectively). This leads us to the follow-ing Γ0(4)-version of the Eichler–Shimura–Manin theorem (see [3,9,11,13]).
Corollary 1.9 Let f∈ Sw+2(Γ0(4)).
(1) If r1(f )= r3(f )= · · · = rw−3(f )= 0, then f = 0. (2) If r2(f )= r4(f )= · · · = rw−2(f )= 0, then f = 0.
The proof of Theorems1.6and1.7will be given in Sect.2. Then in Sect.3, we will deduce Theorem1.1from Theorem1.7.
2 Proofs of Theorems1.6and1.7
In this section, we give proofs for Theorems1.6and1.7. First, we recall 2-adic ordinal of a rational number.
Definition 2.1 For a rational number x, let us express x as
x= 2aq p,
where a, p, q are integers such that (p, q)= 1 and p, q are odd. Then the 2-adic ordinal ord2(x)of x is defined by
ord2(x):= a.
We need the following elementary properties of 2-adic ordinal.
Lemma 2.2 For x, y∈ Q, it holds that
ord2(xy)= ord2(x)+ ord2(y), (2.1)
ord2(x+ y) = ord2(x), if ord2(x) <ord2(y), (2.2) ord2(x+ y) ≥ ord2(x)+ 1, if ord2(x)= ord2(y), (2.3)
ord2(B2n)= −1, if n≥ 1. (2.4)
Proof Proofs of (2.1), (2.2) and (2.3) are straightforward and we omit them. We note that (2.4) follows from the well-known Clausen–von Staudt Theorem on the Bernoulli
numbers (see, e.g., [8]).
Here we recall the polynomial S4,w,n(X)for an integer n with 0 < n < w:
S4,w,n(X)= 4˜nXw ˜n + 1B 0 ˜n+1 1 4X − 1 n+ 1B 0 n+1(X).
k 0
We set
aij:= the coefficient of X2j−1in S4,w,2i(X) (i, j= 1, 2, . . . , w/2 − 1). We will show in Lemma2.4that
det 1≤i≤w/2−1 1≤j≤w/2−1
[aij] = 0. (2.5)
To do so, we need the following lemma:
Lemma 2.3 The 2-adic ordinal ord2(aij)of aij satisfies the following: ord2(ai,i)= −2, for i= 1, 2, . . . , w/2 − 1,
ord2(ai,i+1)= 0, for i= 1, 2, . . . , w/2 − 2,
ord2(ai,i+k)≥ 4(k − 1) + 1, for i = 1, 2, . . . , w/2 − 1; k = 2, 3, . . . , w/2 − 1 − i, ord2(ai,j)≥ −1, for j < i.
Proof We expand S4,w,2i(X)as
S4,w,2i(X)= 4 w−2iXw w− 2i + 1B 0 w−2i+1 1 4X − 1 2i+ 1B 0 2i+1(X) = 1 w− 2i + 1 w−2i+1 =0, even 4 −1 w− 2i + 1 B X2i−1+ − 1 2i+ 1 2i+1 =0, even 2i+ 1 B X2i+1− = 1 w− 2i + 1 w/2 j=i 42j−2i−1 w− 2i + 1 2j− 2i B2j−2iX2j−1 − 1 2i+ 1 i+1 j=1 2i+ 1 2i− 2j + 2 B2i−2j+2X2j−1. Then we know aii= 1 w− 2i + 14 −1 w− 2i + 1 0 B0− 1 2i+ 1 2i+ 1 2 B2, and we have ord2(aii)= ord2(4−1)= −2. We also know
aii+1= 1 w− 2i + 14 1 w− 2i + 1 2 B2− 1 2i+ 1 2i+ 1 0 B0, and we have ord2(aii+1)= ord2(−1/(2i + 1)) = 0.
Now, for aii+kand aij (j < i), we see aii+k= 1 w− 2i + 14 2k−1 w− 2i + 1 2k B2k, aij = − 1 2i+ 1 2i+ 1 2i− 2j + 2 B2i−2j+2.
Hence we have ord2(aii+k)≥ ord2(42k−1/2)= 4k − 3 for k = 2, 3, . . . , w/2 − 1 − i, and ord2(aij)≥ ord2(B2i−2j+2)= −1 for j < i.
This completes the proof.
The following lemma is crucial in our proofs of Theorems1.6and1.7.
Lemma 2.4 Set D= det 1≤i≤w/2−1 1≤j≤w/2−1 [aij]. Then ord2(D)= −w + 2. In particular, we have det 1≤i≤w/2−1 1≤j≤w/2−1 [aij] = 0.
Proof Let us set d= w/2−1, and let id denote the identity element of the symmetric
group Sd of degree d.
From Lemma2.3, we know that
ord2(aii)= −2 and ord2(aij)≥ −1 if i = j. Therefore, for an element σ in Sd, we have
ord2(a1σ (1)a2σ (2)· · · adσ (d))= −2d = −w + 2 if σ = id, ord2(a1σ (1)a2σ (2)· · · adσ (d))≥ −w + 1 if σ= id. Noting that the determinant D is given by
D= σ
ε(σ )a1σ (1)a2σ (2)· · · adσ (d),
where the sum runs over all elements in the permutation group Sd, and ε(σ ) denotes
+1 or −1 according to whether the permutation σ is even or odd, we have
ord2(D)= ord2(a11a22· · · add)= −w + 2.
k 0
Now we are ready to give proofs of Theorems1.6and1.7.
Proofs of Theorems1.6and1.7 In Lemma2.4, we proved that det
1≤i≤w/2−1 1≤j≤w/2−1
[aij] = 0. (2.6)
Since
aij= the coefficient of X2j−1in S4,w,2i(X) (i, j= 1, 2, . . . , w/2 − 1), the inequality (2.6) shows that S4,w,2i (i= 1, 2, . . . , w/2 − 1) are linearly indepen-dent. This implies Theorem1.6.
Next we note that
aij = the coefficient of X2j−1in S4,w,2i(X)
= the coefficient of X2j−1 in r−(RΓ0(4),w,2i)(X) = − w 2j− 1 rw−2j+1(RΓ0(4),w,2i) = − w 2j− 1 cw(RΓ0(4),w,2i, RΓ0(4),w,w−2j+1). Then, from (2.6), we have
w/2−1 j=1 − w 2j− 1 cw det 1≤i≤w/2−1 1≤j≤w/2−1 (RΓ0(4),w,2i, RΓ0(4),w,w−2j+1) = 0.
From this, it follows that det 1≤i≤w/2−1 1≤j≤w/2−1 (RΓ0(4),w,2i, RΓ0(4),w,w−2j+1) = 0. (2.7)
This implies that RΓ0(4),w,2i, i= 1, 2, . . . , w/2 − 1, are linearly independent, and so are RΓ0(4),w,w−2j+1, j= 1, 2, . . . , w/2 − 1. Now taking into account the dimen-sion of Sw+2(Γ0(4)), we conclude that both {RΓ0(4),w,n| n = 2, 4, . . . , w − 2} and
{RΓ0(4),w,n| n = 3, 5, . . . , w − 1} are bases of Sw+2(Γ0(4)). By applying the Atkin– Lehner involution, we know that{RΓ0(4),w,n| n = 1, 3, . . . , w − 3} also form a basis for Sw+2(Γ0(4)). This completes the proof of Theorem1.7.
3 Proof of Theorem1.1and Corollaries1.2and1.3
In the following proposition, a newform in Sk(Γ0(N )) means a normalized Hecke eigenform in the newform subspace of Sk(Γ0(N )). Also, the Petersson inner product of two cusp forms f and g in Sk(Γ0(4)) is defined as (1.5).
Proposition 3.1 Analogue of [6, Proposition 2] Let k≥ 6 be an even integer. For an
integer with 2≤ ≤ k/2 − 2, let E02 and Ek−2 i∞ be the Eisenstein series defined in
(1.2), and set ck, = (k− 2)! (4π )k−1· 4 B2 · 1 1− 22 · 1 1− 22 −kζ (k− 2 ).
(1) If f is a newform in Sk(Γ0(4)), then we have
f, E02 Eki∞−2 = ck, L(f, k− 1)L(f, k − 2 ).
(2) If f is a newform in Sk(Γ0(2)) with f|kW2= ff, then for g(τ )= f (τ) or f (2τ ), we have g, E2 0Eki∞−2 = ck, 1+ f2−k/2 L(f, k− 1)L(g, k − 2 ).
(3) If f is a Hecke eigenform in Sk(SL2(Z)) with T2f = λff, then for g(τ )= f (τ), f (2τ ), or f (4τ ), we have g, E2 0Eki∞−2 = ck, 1+ 2−k+1(1− λf) L(f, k− 1)L(g, k − 2 ).
Moreover, the same formulas hold for = 1 or k/2 − 1 if E02 Eki∞−2 is replaced by
E20(τ )Eki∞−2(τ )− 1 π i(k− 2) d dτE i∞ k−2(τ ), E2i∞(τ )E0k−2(τ )− 1 π i(k− 2) d dτE 0 k−2(τ ), respectively.
Proof The proof follows the argument in [6, Proposition 2], so parts of the proof will be sketchy.
We first consider the case 2≤ < (k − 1)/4. Let f (τ) =anqn∈ Sk(Γ0(4)). According to (1.1), E02 (τ )= 4 B2 (1− 22 ) ∞ n=1 σ2 −1(n)− σ2 −1(n/2) qn= ∞ n=1 e2 (n)qn.
By Rankin’s method, we have
f, E02 Eki∞−2 =(k− 2)! (4π )k−1Lf, (k− 1), (3.1) where Lf, (s)= ∞ n=1 e2 (n)a(n)n−s. (3.2)
k 0
(See [12] and [14, pages 144–146] for more details.) Now assume f (τ ) is a newform in Sk(Γ0(4)). Then
Lf, (s)= 4 B2 (1− 22 ) ∞ n=1 σ2 −1(n)a(n)n−s− ∞ n=1 σ2 −1(n/2)a(n)n−s .
Following the computation in [6, page 822], we find that the first sum above is equal to
L(f, s)L(f, s− 2 + 1) ζ(2)(2s− 2 − k + 2) ,
where ζ(2)(s):= (1 − 2−s)ζ (s). Also, because f is assumed to be a newform on Γ0(4), we have a(2n)= 0 for all n and the second sum above is simply 0. Upon setting s= k − 1, we get the formula in Part (1) for the case 2 ≤ < (k − 1)/4.
We next assume that f is a newform in Sk(Γ0(2)). For the case g= f , aside from a difference in the scalars, the proof is exactly the same as the proof of (i) of Proposition 2 in [6] and we find
Lf, = 4
B2 (1− 22 )
L(f, s)L(f, s− 2 + 1) ζ(2)(2s− 2 − k + 2) ,
from which we obtain the formula in the case g= f . We now consider g(τ) = f (2τ). Letting b = 4 /B2 (1− 22 ), by (3.2), we have Lg, (s)= b ∞ n=1 σ2 −1(n)a(n/2)n−s− b ∞ n=1 σ2 −1(n/2)a(n/2)n−s = 2−sb ∞ n=1 σ2 −1(2n)a(n)n−s− 2−sb ∞ n=1 σ2 −1(n)a(n)n−s. (3.3)
Inserting the identity
σ2 −1(2n)=1+ 22 −1σ2 −1(n)− 22 −1σ2 −1(n/2) into the equation, we obtain
Lg, (s)= 2−s+2 −1b ∞ n=1 σ2 −1(n)a(n)n−s− ∞ n=1 σ2 −1(n/2)a(n)n−s = 2−s+2 −1Lf, (s)= 2−s+2 −1b L(f, s)L(f, s− 2 + 1) ζ(2)(2s− 2 − k + 2) = b L(f, s)L(g, s− 2 + 1) ζ(2)(2s− 2 − k + 2) . (3.4) Setting s= k − 1, we get the formula in Part (2) for the case 2 ≤ < (k − 1)/4.
We now consider the case when f is a normalized Hecke eigenform in Sk(SL2(Z)). Again, when g= f , the proof of the formula is almost the same as the proof of (ii) of
Proposition 2 in [6]. Then when g(τ )= f (2τ), a computation analogous to (3.3) and (3.4) gives us the claimed formula. The proof of the case g(τ )= f (4τ) is similar. This completes the proof of the case 2≤ < (k − 1)/4.
We next consider the case (k+ 1)/4 < ≤ k/2 − 2. Using the fact that the Atkin– Lehner involution W4 is a Hermitian operator with respect to the Petersson inner product, we have
f, E2 0Eki∞−2 =f|W4, Ek0−2 Ei2 ∞
.
When f is a newform in Sk(Γ0(4)) with f|kW4= ff, by the formula in Part (1) with replaced by k/2− , this is equal to
f, E2 0Eki∞−2 = f
f, Ek0−2 Ei2 ∞= fck,k/2− L(f, k− 1)L(f, 2 ). Then from the functional equation
2π √ 4 −s Γ (s)L(f, s)= f(−1)k/2 2π √ 4 −(k−s) Γ (k− s)L(f, k − s) and the identity
ζ (2n)= −(2π i) 2n Γ (2n)
B2n
4n (3.5)
for integers n≥ 1, we get
f, E2 0Eki∞−2 = ck,k/2− L(f, k− 1)L(f, 2 ) = ck, L(f, k− 1)L(f, k − 2 ). Now assume that f is a newform in Sk(Γ0(2)) with f|kW2= ff. Then
(f|kW4)(τ )= (2τ)−kf (−1/4τ) = f(2τ )−k
2√2τkf (2τ )= f2k/2f (2τ ), and consequently, for g(τ )= f (τ),
g, E02 Eki∞−2 = f2k/2
h, E0k−2 E2 i∞
with h(τ )= f (2τ). Applying the formula in Part (2) with replaced by k/2 − , we get g, E02 Eki∞−2 = f2k/2ck,k/2− 1+ f2−k/2 L(f, k− 1)L(h, 2 ) = 2−2 ck,k/2− 1+ f2k/2 L(f, k− 1)L(g, 2 ).
Then from the functional equation for L(f, s) and (3.5), we establish the formula in Part (2) for the case g(τ )= f (τ). The proof of the case g(τ) = f (2τ) is similar.
Now assume that f is a Hecke eigenform in Sk(SL2(Z)) with T2f = λff. We have
(f|kW4)(τ )= (2τ)−kf (−1/4τ) = 2kf (4τ ), and thus, for g(τ )= f (τ),
k 0
with h(τ )= f (4τ). Using the formula in Part (3), we derive that
g, E2 0Eik∞−2 = 2kck,k/2− 1+ 2−k+1(1− λf) L(f, k− 1)L(h, 2 ) = 2k−4 c k,k/2− 1+ 2−k+1(1− λf) L(f, k− 1)L(f, 2 ).
Then, by the functional equation for L(f, s) and (3.5) again, we see that the formula in Part (3) holds for g(τ )= f (τ). The proof of the cases g(τ) = f (2τ) and g(τ) = f (4τ ) is similar. This completes the proof of the formulas for 2≤ ≤ k/2 − 2.
Finally, let us consider the cases = 1 and = k/2 − 1. Assume that = 1. We first recall the well-known transformation formula
E2 aτ+ b cτ+ d = 6 π ic(cτ+ d) + (cτ + d) 2E 2(τ ),
which can be proved easily by considering the logarithmic derivative of the two sides of η((aτ+ b)/(cτ + d))24= (cτ + d)12η(τ )24, where η(τ ) is the Dedekind eta func-tion. It follows that the Eisenstein series E02(τ )= (E2(τ )− E2(2τ ))/3 satisfies
E20 aτ+ b cτ+ d = 1 π ic(cτ+ d) + (cτ + d) 2E0 2(τ ) (3.6)
for alla bc d∈ Γ0(2). Also, since Eki∞−2(τ ) is a modular form of weight k− 2, we have Eki∞−2 aτ+ b cτ+ d = (k − 2)c(cτ + d)k−1Ei∞ k−2(τ )+ (cτ + d)kEki∞−2(τ ). (3.7) Thus, h(τ )= E20Ek−2i∞(τ )− 1 π i(k− 2)E i∞ k−2(τ )
is a cusp form of weight k on Γ0(4). Now we have
Eik∞−2(τ )= γ∈Γ∞\Γ0(4)
1 (cτ+ d)k−2,
where Γ∞is the subgroup generated by1 10 1and for γ ∈ Γ∞\Γ0(4), we write γ=
a b
c d
. It follows that, for f ∈ Sk(Γ0(4)),
(f, h)= γ∈Γ∞\Γ0(4) Γ0(4)\H f (τ ) E20(τ ) (cτ+ d)k−2+ 1 π i c (cτ+ d)k−1 ykdx dy y2 = γ∈Γ∞\Γ0(4) γ (Γ0(4)\H) fγ−1τ × E20(γ−1τ ) (cγ−1τ+ d)k−2+ 1 π i c (cγ−1τ+ d)k−1 ×Im γ−1τkdx dy y2 ,
where we write τ= x + iy. From the transformation formula (3.6), we get E20(γ−1τ ) (cγ−1τ+ d)k−2+ 1 π i c (cγ−1τ + d)k−1= (cτ − a) kE0 2(τ ). Consequently, if f (τ )=a(n)qnand E20(τ )=e2(n)qn, we have
(f, h)= γ∈Γ∞\Γ0(4) γ (Γ0(4)\H) f (τ )E20(τ )yk dx dy y2 = ∞ 0 1 0 ∞ m,n=1
a(m)e2(n)e2π i(n−m)xe−2π(m+n)yyk−2dx dy
=Γ (k− 1) (4π )k−1 ∞ n=1 a(n)e2(n)n−(k−1)= (k− 2)! (4π )k−1Lf,1(k− 1),
and we are back to (3.1). Therefore, the formulas in the statement of the proposition hold if we replace E20Eki∞−2 by h= E20Eki∞−2− Eki∞−2/π i(k− 2). Finally, the case E2i∞E0k−2− E0k−2/π i(k− 2) can be proved by applying the Atkin–Lehner involu-tion, as what we did for the case (k+ 1)/4 < ≤ k/2 − 2. This completes the proof
of the proposition.
We now prove Theorem1.1and Corollaries1.2and1.3.
Proof of Theorem1.1 Let k≥ 6 be an even integer and let
d= dim Sk
Γ0(4)=k 2− 2.
Let h1= E02Eik−2∞ − Ek−2i∞/π i(k− 2) and hj = E2j0 Eik−2j∞ for j= 2, . . . , d. As in Proposition3.1, by a newform in Sk(Γ0(N )), we mean a normalized Hecke eigenform
k 0
in the newform subspace of Sk(Γ0(N )). We first choose a basis for Sk(Γ0(4)) to be
f (τ ), f (2τ ), f (4τ ): f a Hecke eigenform in Sk SL2(Z) ∪ f (τ ), f (2τ ): f a newform in Sk Γ0(2) ∪ f (τ ): f a newform in Sk Γ0(4)
and label the functions by g1, . . . , gd. We also let fi denote the corresponding new-form from which gi originates. Consider the d× d matrix
A=(gi, hj)
i,j=1,...,d
formed by the Petersson inner product of gi and hj. Since {gi} is a basis for Sk(Γ0(4)),{hj} is a basis if and only if det A = 0. Now by the formulas in Proposi-tion3.1, we have det A= d j=1 ck,j d i=1 biL(fi, k− 1) detL(gi, k− 2j) i,j=1,...,d, where bi= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1+ 2−k+1(1+ λfi) if fi is a Hecke eigenform in Sk(Γ (1)) with T2fi= λfifi, 1+ fi2−k/2 if fi is a newform in Sk(Γ0(2)) with fi|kW2= fifi, 1 if fi is a newform in Sk(Γ0(4)).
The numbers ck,j are clearly nonzero. Also, since fi are assumed to be normal-ized Hecke eigenforms, we know that biL(fi, k− 1) = 0. Therefore, to show that det A= 0, it suffices to show that det[L(gi, k− 2j)] = 0.
Now by (1.3), we have L(gi, k− 2j) =(−2πi) k−2j Γ (k− 2j) i∞ 0 gi(τ )τk−2j−1dτ=(−2πi) k−2j Γ (k− 2j) rk−2j−1(gi) =(−2πi)k−2j 2Γ (k− 2j)(2i) k−1(g i, Rk−2j−1),
where Rn= RΓ0(4),k−2,nis the cusp form in Sk(Γ0(4)) characterized by the property (1.4). Thus, det[L(gi, k− 2j)] = 0 if and only if det[(gi, Rk−2j−1)] = 0. However,
{Rk−2j−1}dj=1 is a basis of Sk(Γ0(4)) by Theorem1.7and Remark 1.8, and so is
{gi}di=1by the assumption. Hence we know that det[(gi, Rk−2j−1)] = 0, and we can conclude that the set
E20Eki∞−2− 1 π i(k− 2)E i∞ k−2 ∪ En0Eki∞−n| n = 4, 6, . . . , k − 4
is a basis for Sk(Γ0(4)). Applying the Atkin–Lehner involution to this basis, we see that the other set in the statement of theorem is also a basis.
Proofs of Corollaries1.2 and 1.3 Let W : Sk(Γ0(4))→ Sk(Γ0(4)) be defined by W (f )= f |kW4for any f in Sk(Γ0(4)). Let I denote the identity automorphism of Sk(Γ0(4)). Since W2= I , we have Sk Γ0(4),+= Ker(I − W) = Im(I + W), Sk Γ0(4),− = Ker(I + W) = Im(I − W).
Now, from Theorem1.1, we know
E2i∞Ek0−2− 1 π i(k− 2)E 0 k−2 ∪ Eni∞Ek0−n| n = 4, 6, . . . , k − 4 is a basis for Sk(Γ0(4)). Then the set
E2i∞Ek0−2− 1 π i(k− 2)E 0 k−2 ∪ Eni∞Ek0−n+ En0Eki∞−n| n = 4, 6, . . . , 2k/4 ∪ Ein∞E0k−n| n = 2k/4 + 2, 2k/4 + 4, . . . , k − 4 is also a basis for Sk(Γ0(4)). In particular,
Eni∞Ek0−n+ En0Eki∞−n| n = 4, 6, . . . , 2k/4
is linearly independent. Furthermore, since Eni∞Ek0−n+ En0Eki∞−n∈ Sk(Γ0(4),+) and dim Sk(Γ0(4),+) = k4 − 1, we know
Eni∞Ek0−n+ En0Eki∞−n| n = 4, 6, . . . , 2k/4 is a basis for Sk(Γ0(4),+).
Next, from Theorem1.1, we know that Sk(Γ0(4),−) = Im(I − W) is spanned by
E2i∞Ek0−2− E20Eki∞−2− 1 π i(k− 2) E0k−2− Eik∞−2 ∪ Eni∞E0k−n− En0Eki∞−n| n = 4, 6, . . . , k − 4. Then Sk(Γ0(4),−) is also spanned by the set
E2i∞Ek0−2− E20Eki∞−2− 1 π i(k− 2) Ek0−2− Eik∞−2 ∪ Eni∞Ek0−n− En0Eki∞−n| n = 4, 6, . . . , k − 2k/4 − 2.
Now, noting that dim Sk(Γ0(4),−) = k/2 − k/4 − 1, we conclude the set above is
a basis of Sk(Γ0(4),−).
References
1. Chan, H.H., Chua, K.S.: Representations of integers as sums of 32 squares. Ramanujan J. 7, 79–89 (2003)
k 0
2. Chan, H.H., Krattenthaler, C.: Recent progress in the study of representations of integers as sums of squares. Bull. Lond. Math. Soc. 37, 818–826 (2005)
3. Eichler, M.: Eine Verallgemeinerung der Abelschen Integrale. Math. Z. 67, 267–298 (1957) 4. Fukuhara, S., Yang, Y.: Period polynomials and explicit formulas for Hecke operators on Γ0(2). Math.
Proc. Camb. Philos. Soc. 146(2), 321–350 (2009)
5. Grosswald, E.: Representations of Integers as Sums of Squares. Springer, New York (1985) 6. Imamo¯glu, Ö., Kohnen, W.: Representations of integers as sums of an even number of squares. Math.
Ann. 333(4), 815–829 (2005)
7. Kilgor, K.: Die Vermutung von Chan–Chua und die Kombinatorik von Windungselementen. Ph.D. thesis, Heidelberg (2010)
8. Koecher, M.: Klassische elementare Analysis. Birkhäuser, Basel (1987)
9. Kohnen, W., Zagier, D.: Modular forms with rational periods. In: Modular Forms, Durham, 1983. Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., pp. 197–249. Horwood, Chichester (1984) 10. Long, L., Yang, Y.: A short proof of Milne’s formulas for sums of integer squares. Int. J. Number
Theory 1(4), 533–551 (2005)
11. Manin, Yu.I.: Periods of cusp forms, and p-adic Hecke series. Mat. Sb. 92(134), 378–401 (1973) 12. Rankin, R.A.: The scalar product of modular forms. Proc. Lond. Math. Soc. 2, 198–217 (1952) 13. Shimura, G.: Sur les intégrales attachées aux formes automorphes. J. Math. Soc. Jpn. 11, 291–311
(1959)
14. Zagier, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. In: Modular Functions of One Variable, VI. Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976. Lec-ture Notes in Math., vol. 627, pp. 105–169. Springer, Berlin (1977)
