OF SQUARES
SHINJI FUKUHARA AND YIFAN YANG
ABSTRACT. In this paper, we find a basis for the space Sk(Γ0(4)) of modular 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 sums of 4s integral squares.
1. INTRODUCTION AND STATEMENTS OF RESULTS
Throughout the paper we assume that k is an even positive integer. Let Γ be a congru-ence 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. For Sk(Γ0(2)), we have given in [2] a basis {E2ji∞E
0
k−2j | j = 2, . . . , d + 1}, where
Eni∞and En0normalized Eisenstein series of weight n for the cusps i∞ and 0, respectively, and d = dim Sk(Γ0(2)) = bk/4c − 1. The existence of such a basis was suggested by [4].
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 nonnegative integer n as sums of s integral
squares.
To state our main results, let us first recall that the group Γ0(4) has 3 cusps, represented
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 ) := X d/c∼α 1 (cτ − d)k, Date: November 22, 2010.
2000 Mathematics Subject Classification. Primary 11F11; Secondary 11F30, 11F67.
Key words and phrases. modular forms (one variable), period polynomials, Fourier coefficients of modular forms, sum of squares.
The first author was partially supported by Grant-in-Aid for Scientific Research (No. 22540096), Japan Soci-ety for the Promotion of Science. The second author was partially supported by Grant 99 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.
where the sum runs over all cusps d/c equivalent to α under Γ0(4). More explicitly, we have Ei∞ k (τ ) = 1 2 X (c,d)=1,4|c 1 (cτ − d)k = 1 2k− 1 2 kE k(4τ ) − Ek(2τ ) , E0 k(τ ) = 1 2 X (c,d)=(c,4)=1 1 (cτ − d)k = 2k 2k− 1(Ek(τ ) − Ek(2τ )) Ek1/2(τ ) =1 2 X (c,d)=1,2|c,4-c 1 (cτ − d)k = 1 2k− 1 −Ek(τ ) + (2 k+ 1)E k(2τ ) − 2kEk(4τ ) , (1.1) where Ek(τ ) = 1 + (2πi)k Γ(k)ζ(k) ∞ X n=1 σk−1(n)qn= 1 − 2k Bk ∞ X n=1 σk−1(n)qn, q = e2πiτ,
is the Eisenstein series of weight k on SL2(Z) and Bkis the kth Bernoulli number. (See
Lemma 3.2 of [7] 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 [7] are essentially our Eki∞, E
0 k,
and Ek1/2here, respectively. This is because Γ0(4) is conjugate to Γ(2) by (2 00 1).) These
Eisenstein series have the property that lim τ →i∞(E α k kγ)(τ ) = ( 1, if a/c ∼ α, 0, if a/c 6∼ α, for all γ = a b c d ∈ SL2(Z). In particular, if α 6∼ β, then E α k1E β
k2is 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
(1.2) Ei∞k (τ ) = E i∞ k (τ ), E 0 k(τ ) = (E i∞ k kW4)(τ ) = 2−kEk0(τ ),
where W4denotes the Atkin-Lehner involution on Mk(Γ0(4)). In addition, for k = 2, we
also define the Eisenstein series E2α(τ ) using the Fourier expansions given in (1.1). These
Eisenstein series E2α(τ ) are not a 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 πikf 0(τ ), E0 2(τ )f (τ ) − 1 πikf 0(τ )
are modular forms of weight k + 2 on Γ0(4). (See Equations (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 {Ei∞ 2 E 0 k−2− 1 πi(k − 2)E 0 k−2 0 }[{Ei∞ n E 0 k−n| n = 4, 6, . . . , k − 4} and {E0 2Ek−2i∞ − 1 πi(k − 2)E i∞ k−2 0 }[{E0 nEi∞k−n| n = 4, 6, . . . , k − 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 sums of s integral
squares. (See [3] for a survey 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= X n∈Z qn2 !s , q = e2πiτ. When s is even, we have
Θ(γτ )s= −1 d s/2 (cτ + d)s/2Θ(τ )s for all γ = a b c d
∈ Γ0(4), where −1d is the Legendre symbol. Thus, for a positive
integer s, the function Θ(τ )4sis a linear combination of Eisenstein series E2si∞(τ ), E2s0(τ ),
E2s1/2(τ ), and the functions in Theorem 1.1. In fact, we can do a little better. The theta function Θ(τ ) satisfies
Θ −1 4τ = r 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, 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∈ C Ei∞ 2s (τ ) + (−1) sE0 2s(τ ) M S2s(Γ0(4), (−1)s). Now we have dim Sk(Γ0(4), +) = k 4 − 1, dim Sk(Γ0(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 be an even integer, then {Eni∞E 0 k−n+ E 0 nE i∞ k−n| n = 4, 6, . . . , 2 bk/4c}
is a basis for Sk(Γ0(4), +). In particular, if k ≡ 0 mod 4, then Θ(τ )2k is a linear
combination ofEi∞
k (τ ) + Ek0(τ ) and the functions above.
Corollary 1.3. If k ≥ 6 is an even integer, then {Ei∞ 2 E 0 k−2− E 0 2E i∞ k−2− 1 πi(k − 2)(E 0 k−2 0 − Ei∞ k−2 0 )} [ {Ei∞ n Ek−n0 − En0Ek−ni∞ | n = 4, 6, . . . , k − 2 bk/4c − 2}
is a basis for Sk(Γ0(4), −). In particular, if k ≡ 2 mod 4, then Θ(τ )2k is a linear
combination ofEi∞ k (τ ) − E
0
We remark that since Γ+0(4) is conjugate to Γ0(2) by 1 1/20 1 , we can first obtain a
basis for Sk(Γ0(2)) and apply τ 7→ 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 [4]. However,
this method only work for the cases 8|s. Also, we note that the basis for Sk(Γ0(4), +)
obtained in this way is different from the basis in Corollary 1.2.
Example 1.4. Here we give some formulas for r4s(n). In the following, we let
fs,0:= E2si∞+ (−1)sE2s0, fs,2:= E2i∞E 0 2s−2+ (−1) sE0 2E i∞ 2s−2− 1 πi(2s − 2)(E 0 2s−2 0 + (−1)sE2s−2i∞ 0), fs,n:= Eni∞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 737235 f7,4+ 460309 245745f7,6, Θ32= f8,0+ 11379631232 4392213525f8,4− 13142016 6506983f8,6+ 967923424 627459075f8,8, Θ36= f9,0+ 929569 3202291f9,2− 2997123429668 1165073523075f9,4+ 817033178804 317747324475f9,6 −130045826398 35305258275 f9,8.
We now indicate how Theorem 1.1 is proved. We shall see that Theorem 1.1 is 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
(1.3) rn(f ) :=
Z i∞
0
f (z)zndz
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
(1.4) rn(f ) = cw(f, RΓ0(N ),w,n), cw:= 2
−1(2i)w+1
for all cusp forms f of the same weight on Γ0(N ). Here
(1.5) (f, g) :=
Z Z
Γ0(N )\H
f (z)g(z)ywdx dy, z = x + iy,
denotes the Petersson inner product of f and g. We now explain the relation between RΓ0(4),w,nand E
i∞ n Ek−n0 .
Using Rankin’s method [9] and following the argument in the proof of Proposition 2 of [4], 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, Eni∞Ek−n0 ) = ck,nL(f, k − 1)L(f, n),
where L(f, s) denotes the L-function associated to f and ck,nis a constant depending on k
and n. (See Proposition 3.1 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) = c 0
k,nL(f, n + 1)
for some constant c0k,nindependent of f . Therefore, even though Eni∞Ek−n0 is not
pre-cisely a multiple of RΓ0(4),w,n−1, we can still deduce linear independence among E i∞ n Ek−n0
from 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) := Z 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,nare computed in [2] and will be crucial in our proof
of Theorem 1.1. To state the formula, we let Bm(x) (resp. Bm) denote the mth Bernoulli
polynomial (resp. number). By B0
m(x), we denote the mth Bernoulli polynomial without
its B1-term ([6, p. 208]): Bm0(x) := X 0≤i≤m i6=1 m i Bixm−i= X 0≤i≤m i even 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) := Nn˜Xw ˜ n + 1 B 0 ˜ 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 [2].
Theorem 1.5 ([2, Theorem 1.1]). Let N be an integer greater than 1. For an even integer n with 0 < n < w, we have
r−(RΓ0(N ),w,n)(X) = SN,w,n(X).
Also, for an odd integern 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 Y p|N 1 − p−(n+1) 1 − p−(w+2) − 1 Nn+1 Y p|N 1 − p−(˜n+1) 1 − p−(w+2) , wherep runs over all prime divisors of N .
In the sequel, we focus on the case N = 4. Furthermore we consider only vector spaces over C, 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 [2], where explicit evaluation
of Hankel determinants formed by Bernoulli numbers is the main ingredient. Here the key to our proof of Theorem 1.6 is 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 Theorem 1.6, we can derive the following result. Theorem 1.7. (1)
{RΓ0(4),w,n| n = 1, 3, . . . , w − 3}
form a basis forSw+2(Γ0(4)).
(2)
{RΓ0(4),w,n| n = 2, 4, . . . , w − 2}
form a basis forSw+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 [2, p. 330] for Atkin-Lehner involution WN. In Theorem 1.7 (1), the basis can be
replaced by
{RΓ0(4),w,n| n = 3, 5, . . . , w − 1},
deleting n = 1 and adding n = w − 1. These correspond each other by 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 lead us to the following Γ0(4)-version of the Eichler-Shimura-Manin theorem ([1], [6], [8], [10]).
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 Theorems 1.6 and 1.7 will be given in Section 2. Then in Section 3, we will deduce Theorem 1.1 from Theorem 1.7.
2. PROOFS OFTHEOREMS1.6AND1.7
In this section we give proofs for Theorems 1.6 and 1.7. 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
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. [5]).
Here we recall the polynomial S4,w,n(X) for an integer n with 0 < n < w:
S4,w,n(X) = 4n˜Xw ˜ n + 1B 0 ˜ n+1 1 4X − 1 n + 1B 0 n+1(X). We set
aij := the coefficient of X2j−1in S4,w,2i(X) (i, j = 1, 2, . . . , w/2 − 1).
We will show in Lemma 2.4 that
(2.5) det
1≤i≤w/2−1 1≤j≤w/2−1
[aij] 6= 0.
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) = 4w−2iXw w − 2i + 1B 0 w−2i+1( 1 4X) − 1 2i + 1B 0 2i+1(X) = 1 w − 2i + 1 w−2i+1 X `=0, ` even 4`−1w − 2i + 1 ` B`X2i−1+` − 1 2i + 1 2i+1 X `=0, ` even 2i + 1 ` B`X2i+1−` = 1 w − 2i + 1 w/2 X j=i 42j−2i−1w − 2i + 1 2j − 2i B2j−2iX2j−1 − 1 2i + 1 i+1 X j=1 2i + 1 2i − 2j + 2 B2i−2j+2X2j−1.
Then we know aii= 1 w − 2i + 14 −1w − 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 1w − 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−1w − 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 Theorems 1.6 and 1.7. 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] 6= 0.
Proof. Let us set d = w/2 − 1, and let id denote the identity element of the symmetric group Sdof degree d.
From Lemma 2.3, we know that
ord2(aii) = −2 and ord2(aij) ≥ −1 if i 6= 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 6= id.
Noting that the determinant D is given by
D =X
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.
This proves the lemma.
Proofs of Theorems 1.6 and 1.7. In Lemma 2.4, we proved that (2.6) det 1≤i≤w/2−1 1≤j≤w/2−1 [aij] 6= 0. 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 independent.
This implies Theorem 1.6. Next we note that
aij =the coefficient of X2j−1in S4,w,2i(X)
=the coefficient of X2j−1in 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 Y 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) 6= 0.
From this, it follows that
(2.7) det
1≤i≤w/2−1 1≤j≤w/2−1
(RΓ0(4),w,2i, RΓ0(4),w,w−2j+1) 6= 0.
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 dimension 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 {RΓ0(4),w,n| n = 1, 3, . . . , w − 3} also form a basis for Sw+2(Γ0(4)). This
com-pletes the proof of Theorem 1.7.
3. PROOF OFTHEOREM1.1ANDCOROLLARIES1.2AND1.3
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 [4, Proposition 2]). Let k ≥ 6 be an even integer. For a integer` with 2 ≤ ` ≤ k/2 − 2, let E0
2`andE i∞
k−2`be the Eisenstein series defined in(1.2),
and set ck,`= (k − 2)! (4π)k−1 · 4` B2` · 1 1 − 22` · 1 (1 − 22`−k)ζ(k − 2`).
(1) If f is a newform in Sk(Γ0(4)), then we have
(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`0Ek−2`i∞ ) = 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, E02`Ek−2`i∞ ) = 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 E0
2`Ek−2`i∞ is replaced by E20(τ )Ek−2i∞(τ ) − 1 πi(k − 2) d dτE i∞ k−2(τ ), E i∞ 2 (τ )E 0 k−2(τ ) − 1 πi(k − 2) d dτE 0 k−2(τ ), respectively.
Proof. The proof follows the argument in [4, Proposition 2], so parts of the proof will be sketchy.
We first consider the case 2 ≤ ` < (k − 1)/4. Let f (τ ) = P anqn ∈ Sk(Γ0(4)).
According to (1.1), E2`0(τ ) = 4` B2`(1 − 22`) ∞ X n=1 (σ2`−1(n) − σ2`−1(n/2)) qn= ∞ X n=1 e2`(n)qn.
By Rankin’s method, we have
(3.1) (f, E2`0Ek−2`i∞ ) = (k − 2)! (4π)k−1Lf,`(k − 1), where (3.2) Lf,`(s) = ∞ X n=1 e2`(n)a(n)n−s.
(See [9] and [11, Pages 144–146] for more details.) Now assume f (τ ) is a newform in Sk(Γ0(4)). Then
Lf,`(s) = 4` B2`(1 − 22`) ∞ X n=1 σ2`−1(n)a(n)n−s− ∞ X n=1 σ2`−1(n/2)a(n)n−s ! . Following the computation in [4, Page 822], we find 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 [4] 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` ∞ X n=1 σ2`−1(n)a(n/2)n−s− b` ∞ X n=1 σ2`−1(n/2)a(n/2)n−s = 2−sb` ∞ X n=1 σ2`−1(2n)a(n)n−s− 2−sb` ∞ X n=1 σ2`−1(n)a(n)n−s. (3.3)
Inserting the identity
σ2`−1(2n) = (1 + 22`−1)σ2`−1(n) − 22`−1σ2`−1(n/2)
into the equation, we obtain Lg,`(s) = 2−s+2`−1b` ∞ X n=1 σ2`−1(n)a(n)n−s− ∞ X 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 [4]. 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 W4is a Hermitian operator with respect to the Petersson inner product,
we have
(f, E02`Ek−2`i∞ ) = (f W4, Ek−2`0 E i∞ 2` )
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`0Ek−2`i∞ ) = f(f, Ek−2`0 E
i∞
2` ) = 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
(3.5) ζ(2n) = −(2πi)
2n
Γ(2n) B2n
4n for integers n ≥ 1, we get
(f, E2`0Ek−2`i∞ ) = 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 √ 2τ )kf (2τ ) = f2k/2f (2τ )
and consequently, for g(τ ) = f (τ ),
(g, E2`0Ek−2`i∞ ) = f2k/2(h, Ek−2`0 E i∞ 2` )
with h(τ ) = f (2τ ). Applying the formula in Part (2) with ` replaced by k/2 − `, we get (g, E02`Ek−2`i∞ ) = 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 ),
(g, E2`0Ek−2`i∞ ) = 2k(h, Ek−2`0 E2`i∞)
with h(τ ) = f (4τ ). Using the formula in Part (3), we derive that
(g, E2`0Ek−2`i∞ ) = 2kck,k/2−`(1 + 2−k+1(1 − λf))L(f, k − 1)L(h, 2`)
= 2k−4`ck,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 πic(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 function. It
follows that the Eisenstein series E0
2(τ ) = (E2(τ ) − E2(2τ ))/3 satisfies (3.6) E20 aτ + b cτ + d = 1 πic(cτ + d) + (cτ + d) 2E0 2(τ ) for all a b c d ∈ Γ0(2). Also, since E i∞
k−2(τ ) is a modular form of weight k − 2, we have
(3.7) Ek−2i∞ 0 aτ + b cτ + d = (k − 2)c(cτ + d)k−1Ei∞k−2(τ ) + (cτ + d)kEk−2i∞ (τ ). Thus, h(τ ) = E20Ek−2i∞ (τ ) − 1 πi(k − 2)E i∞ k−2 0 (τ ) is a cusp form of weight k on Γ0(4). Now we have
Ek−2i∞(τ ) = X
γ∈Γ∞\Γ0(4)
1 (cτ + d)k−2,
where Γ∞is the subgroup generated by (1 10 1) and for γ ∈ Γ∞\Γ0(4), we write γ = a bc d.
It follows that, for f ∈ Sk(Γ0(4)),
(f, h) = X γ∈Γ∞\Γ0(4) Z Z Γ0(4)\H f (τ ) E0 2(τ ) (cτ + d)k−2 + 1 πi c (cτ + d)k−1 yk dxdy y2 = X γ∈Γ∞\Γ0(4) Z Z γ(Γ0(4)\H) f (γ−1τ ) E0 2(γ−1τ ) (cγ−1τ + d)k−2 + 1 πi c (cγ−1τ + d)k−1 × (Im γ−1τ )k dxdy y2 ,
where we write τ = x + iy. From the transformation formula (3.6), we get E0 2(γ−1τ ) (cγ−1τ + d)k−2 + 1 πi c (cγ−1τ + d)k−1 = (cτ − a) kE0 2(τ ).
Consequently, if f (τ ) =P a(n)qnand E0
2(τ ) =P e2(n)qn, we have (f, h) = X γ∈Γ∞\Γ0(4) Z Z γ(Γ0(4)\H) f (τ )E20(τ )yk dxdy y2 = Z ∞ 0 Z 1 0 ∞ X m,n=1
a(m)e2(n)e2πi(n−m)xe−2π(m+n)yyk−2dx dy
=Γ(k − 1) (4π)k−1 ∞ X 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 E20Ek−2i∞ by h = E20Ek−2i∞ − Ei∞
k−2 0
/πi(k − 2). Finally, the case E2i∞E 0 k−2−
E0 k−2
0
/πi(k − 2) can be proved by applying the Atkin-Lehner involution, as what we did for the case (k + 1)/4 < ` ≤ k/2 − 2. This completes the proof of the proposition.
We now prove Theorem 1.1 and Corollaries 1.2 and 1.3. Proof of Theorem 1.1. Let k ≥ 6 be an even integer and let
d = dim Sk(Γ0(4)) =
k 2 − 2. Let h1 = E20Ek−2i∞ − Ek−2i∞
0
/πi(k − 2) and hj = E02jEk−2ji∞ for j = 2, . . . , d. As in
Proposition 3.1, by a newform in Sk(Γ0(N )), we mean a normalized Hecke eigenform 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 fidenote the corresponding newform
from which gioriginates. Consider the d × d matrix
formed by the Petersson inner product of giand hj. Since {gi} is a basis for Sk(Γ0(4)),
{hj} is a basis if and only if det A 6= 0. Now by the formulas in Proposition 3.1, we have
det A = d Y j=1 ck,j d Y i=1 biL(fi, k − 1) ! det[L(gi, k − 2j)]i,j=1,...,d, where bi =
1 + 2−k+1(1 + λfi), if fiis a Hecke eigenform in Sk(Γ(1)) with T2fi= λfifi,
1 + fi2 −k/2, if f iis a newform in Sk(Γ0(2)) with fi kW2= fifi, 1, if fiis a newform in Sk(Γ0(4)).
The numbers ck,jare clearly nonzero. Also, since fiare assumed to be normalized Hecke
eigenforms, we know that biL(fi, k − 1) 6= 0. Therefore, to show that det A 6= 0, it
suffices to show that det[L(gi, k − 2j)] 6= 0.
Now by (1.3), we have L(gi, k − 2j) = (−2πi)k−2j Γ(k − 2j) Z 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 (gi, Rk−2j−1),
where Rn = RΓ0(4),k−2,n is the cusp form in Sk(Γ0(4)) characterized by the property
(1.4). Thus, det[L(gi, k − 2j)] 6= 0 if and only if det[(gi, Rk−2j−1)] 6= 0. However,
{Rk−2j−1}dj=1is a basis of Sk(Γ0(4)) by Theorem 1.7 and Remark 1.8, and so is {gi}di=1
by the assumption. Hence we know that det[(gi, Rk−2j−1)] 6= 0, and we can conclude that
the set {E0 2E i∞ k−2− 1 πi(k − 2)E i∞ k−2 0 } ∪ {E0 nE i∞ k−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 Corollaries 1.2 and 1.3. Let W : Sk(Γ0(4)) → Sk(Γ0(4)) be defined by W (f ) =
f |kW4 for 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 Theorem 1.1, we know {E2i∞E 0 k−2− 1 πi(k − 2)E 0 k−2 0 }[{Eni∞E 0 k−n| n = 4, 6, . . . , k − 4}
is a basis for Sk(Γ0(4)). Then the set
{Ei∞ 2 E 0 k−2− 1 πi(k − 2)E 0 k−2 0 }[{Ei∞ n E 0 k−n+ E 0 nE i∞ k−n| n = 4, 6, . . . , 2 bk/4c} [ {Eni∞E 0 k−n| n = 2 bk/4c + 2, 2 bk/4c + 4, . . . , k − 4}
is also a basis for Sk(Γ0(4)). In particular,
{Eni∞E 0 k−n+ E 0 nE i∞ k−n| n = 4, 6, . . . , 2 bk/4c}
is linearly independent. Furthermore, since Eni∞Ek−n0 + E 0 nEk−ni∞ ∈ Sk(Γ0(4), +) and dim Sk(Γ0(4), +) = k 4 − 1, we know {Ei∞ n E 0 k−n+ E 0 nE i∞ k−n| n = 4, 6, . . . , 2 bk/4c} is a basis for Sk(Γ0(4), +).
Next, from Theorem 1.1, we know that Sk(Γ0(4), −) = Im (I − W ) is spanned by
{Ei∞ 2 E 0 k−2− E 0 2E i∞ k−2− 1 πi(k − 2)(E 0 k−2 0 − Ei∞ k−2 0 )} [ {Ei∞ n E 0 k−n− E 0 nE i∞ k−n| n = 4, 6, . . . , k − 4}.
Then Sk(Γ0(4), −) is also spanned by the set
{E2i∞E 0 k−2− E 0 2E i∞ k−2− 1 πi(k − 2)(E 0 k−2 0 − Ek−2i∞ 0 )} [ {Ei∞ n E 0 k−n− E 0 nE i∞ k−n| n = 4, 6, . . . , k − 2 bk/4c − 2}.
Now, noting that dim Sk(Γ0(4), −) = k/2 − bk/4c − 1, we conclude the set above is a
basis of Sk(Γ0(4), −).
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DEPARTMENT OFMATHEMATICS, TSUDACOLLEGE, TSUDA-MACHI2-1-1, KODAIRA-SHI, TOKYO 187-8577, JAPAN
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DEPARTMENT OFAPPLIEDMATHEMATICS, NATIONALCHIAOTUNGUNIVERSITY ANDNATIONALCEN -TER FORTHEORETICALSCIENCES, HSINCHU, TAIWAN300
