Math. Z. 250, 257–266 (2005)
Mathematische Zeitschrift
DOI: 10.1007/s00209-004-0740-2
On the Kohnen-Zagier formula in the case of level 4pm Hiroshi Sakata Waseda University Senior High School, Kamisyakujii 3-31-1, Nerima-ku, Tokyo, 177-0044, Japan (e-mail:
[email protected]) Received: 14 February 2003; in final form: 1 April 2004 / Published online: 7 January 2005 – © Springer-Verlag 2005
Mathematics Subject Classification (2000): 11F30, 11F37, 11F67
1. Introduction Let g be a Hecke eigen newform in Kohnen’s space of cusp forms of half-integral weight k + 1/2 and level 4N with N an odd number, and let f be the primitive form of weight 2k and level N associated to g by the Shimura correspondence. When N is squarefree, it was found by Kohnen-Zagier [5] and Kohnen [3] that the square of cg (|D|) of the |D|-th Fourier coefficient of g for a fundamental discriminant D with (−1)k D > 0, (N, D) = 1 is expressed explicitly as the product of the central value L(f, D, k) of the L-function of f twisted by a quadratic character D with some elementary factors. Kohnen-Zagier’s results have been generalized in several directions. Namely by Kohnen [4] to the case of primitive form f with arbitrary odd level, by Shimura [8] to the case of Hilbert modular form g of half integral weight over totally real number fields. It should be noted that these results are all obtained under the condition that Kohnen’s spaces have the multiplicity one theorem. On the other hand, Ueda [9, 10] showed that Kohnen’s spaces do not necessarily satisfy a multiplicity one theorem for general N by using the twisting operators and trace formula of them. This fact causes a serious difficulty when one wants to find an analogous result in the case g has an arbitrary level. Concerning this problem, Kojima [6] obtained a result when Kohnen’s spaces have multiplicity two, in the case of primitive form f with squarefree odd level and general character by embedding Kohnen’s space into the space of Hilbert modular forms of half-integral weight and developing Shimura’s method on the latter. Kojima [6]’s work also shows that Fourier coefficients of Hecke eigen newforms in Kohnen’s spaces which do not have multiplicity one theorem have closely related to central values of twisted automorphic L-function.
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The purpose of this paper is to give a result of Kohnen-Zagier type in the case of a primitive form f of odd prime power level p m by using Ueda’s result [9–12] . In this case, Kohnen’s space does not necessarily have multiplicity one. Therefore, this result is not included in Kohnen [4] and Kojima [6] .
2. Notations Throughout this paper, we use the following notations. For z in the upper half plane H, we put e(z) = e2πiz . The symbol
c
defined d for c, d ∈ Z is the quadratic residue symbol defined by Shimura [7]. We let Γ0 (N ) = αβ ∈ SL2 (Z) γ ≡ 0 (mod N ) by a natural number N . γ δ
3. Twisting operators and the decomposition of Kohnen’s space Let k be a positive integer. Suppose that M is a natural number. By S2k (M) we denote the space of all holomorphic cusp forms of weight 2k with the trivial character on the congruence subgroup Γ0 (M). We also denote the space of cusp forms of weight k + 1/2 with the Dirichlet character χ on Γ0 (M) by Sk+1/2 (N, χ ) when M = 4N and N is a natural number. Especially, if χ is the trivial character, we denote Sk+1/2 (N, χ ) by Sk+1/2 (N ). If N is an odd natural number, we define the K Kohnen’s space Sk+1/2 (N ) as follows; K (N ) = Sk+1/2
g(z) =
cg (n)e(nz) ∈ Sk+1/2 (N )
n≥1
cg (n) = 0
for (−1) n ≡ 2, 3 k
(mod 4) .
This space is the Hilbert space with respect to the Petersson inner product dxdy 1 f (τ )g(τ )y κ 2 (Re(τ ) = x, Im(τ ) = y), f, g = [Γ0 (1) : Γ ] Γ \H y where f and g are cusp forms of weight κ ∈ 21 Z on some subgroup Γ of finite index in Γ0 (1). As well known result, we have the theory of newforms for S2k (M), new (M) what is spanned by newi.e. we have the multiplicity one theorem for S2k forms of S2k (M). Furthermore Kohnen [2, 3] established the theory of newforms K for Sk+1/2 (N ) when N is odd squarefree, i.e. Theorem 1 (Kohnen [2, 3]). The notation being as above, there is the subspace K K Sk+1/2 (N )new in Sk+1/2 (N ) such that new K (N )new ∼ (N ) Sk+1/2 = S2k
(as H ecke modules).
On the Kohnen-Zagier formula in the case of level 4p m
259
On the other hand, Kohnen’s result does not work when N is not squarefree. Because K there is a case such that all common eigen subspace of Sk+1/2 (N ) for Hecke operators have dimension ≥ 2 and hence the multiplicity one theorem does not hold K good. Ueda succeeded in resolving this difficulty by decomposing Sk+1/2 (N ) into eigen subspaces of twisting operator
n cg (n)e(nτ ), g|RI (τ ) = lI n≥1 (−1)k n≡0,1 (mod 4)
whereI is any subset of = p|p is an odd prime number such that p 2 |N and lI = p∈I p. Especially we have the following theorem in the case of N = p m . Theorem 2 (Ueda [10]). Let k be a natural number with k ≥ 2, m be a natural number with m ≥ 3, and p be an odd prime number. Then we have K,0 K,+1 K,−1 K Sk+1/2 (p m ) = Sk+1/2 (p m ) ⊕ Sk+1/2 (p m ) ⊕ Sk+1/2 (p m ) K,±1 K where Sk+1/2 (p m ) is the ±1-eigen subspace of Sk+1/2 (p m ) on the twisting operator Rp . Moreover we have p K,0 K K Sk+1/2 (p m ) = Ker(Rp |Sk+1/2 p m−1 , , (p m )) = g(pτ ) g(τ ) ∈ Sk+1/2 K,0 K (in other words, Sk+1/2 (p m ) is the space of ‘oldforms’ in Sk+1/2 (p m )) and new,+1 m S2k (p ), if m is even, k K,±1 m new ∼ Sk+1/2 (p ) −1 = new,± p S2k (p m ), if m is odd. K,±1 as Hecke modules, where Sk+1/2 (p m )new is the orthogonal complement of K,±1 K,±1 new,±1 m Sk+1/2 (p m−1 ) in Sk+1/2 (p m ) and S2k (p ) is the ±1-eigen subspace of new m S2k (p ) on the Atkin-Lehner involution W (p m ). From this isomorphism, we have K,±1 the multiplicity one theorem for the space Sk+1/2 (p m )new .
Remark 1. The case of m = 2 is more complicated in structure. Actually, Rp can new (p l ) (l = 0, 1) to ones in the S new (p 2 ), and consealways map forms in S2k 2k K quently there is a case such that Sk+1/2 (p 2 )new has the component corresponding new (p)|R ⊕S (1)|R in addition to S new (p 2 ) by the Shimura correspondence, to S2k p 2k p 2k i.e. Theorem 3 (Ueda [12]). Let k and p be the same as above. Then we have the following K,+ K,− Sk+1/2 (p 2 )new ∼ = Sk+1/2 (p 2 )new
new
−1 1 ∗,+ ∼ 1+ S2k (p)|Rp ⊕ S2k (1)|Rp , = S2k (p 2 ) ⊕ 2 p
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∗ (p 2 ) is the orthogonal complement of S new (p)|R ⊕ S (1)|R in where S2k p 2k p 2k ∗,+ 2 ∗ (p 2 ) 2 new S2k (p ) with the Petersson inner product and S2k (p ) is a subspace of S2k as ∗,+ 2 ∗ (p ) = f ∈ S2k (p 2 )f |W (p 2 ) = f . S2k
4. Periods of cusp forms and the Shimura correspondence Let k ≥ 2, N an odd natural number, m ≥ 1 with (−1)k m ≡ 0, 1 (mod 4), and D be a (fundamental) discriminant with (−1)k D > 0 and (N, D) = 1. As usual the group SL2 (Z) acts on the set consisting of integral binary quadratic forms Q = [a, b, c](X, Y ) = aX2 + bXY + cY 2 by [a, b, c] ◦
αβ γ δ
(X, Y ) = [a, b, c](αX + βY, γ X + δY )
preserving the discriminant = b2 − 4ac, the number of classes with fixed values of this invariant being finite. We consider the classification with respect to the subgroup Γ0 (N ). A further invariant in this case is the greatest common divisor of a and N , which we suppose to be N . Therefore we set the finite set QN,|D|m of Γ0 (N )-inequivalent integral binary quadratic forms of discriminant |D|m and with N |a. Nextly, we define the period integral of f ∈ S2k (N ) (associated to a quadratic form Q in QN,|D|m ) by rk,N,Q (f ) =
f (z)Q(z, 1)k−1 dz, γQ
where γQ is a geodesic integral determined by Q, which is the image in Γ0 (N )\H of the semicircle az2 + bRe(z) + c = 0 oriented from left to right if a > 0, from −c right to left if a < 0, from −c b to i∞ if a = 0 and b > 0, and from i∞ to b if a = 0 and b < 0. We also define the periods of f
rk,N (f ; D, (−1)k m) =
ωD (Q)rk,N,Q (f ),
Q∈QN,|D|m
where ωD is the genus character given in Kohnen [3]. Lastly we define the D-th K (N ) to S2k (N ) defined by Shimura correspondence of Sk+1/2 g|Sk,N,D (τ ) =
n≥1
d|N,(d,N)=1
D d
d
k−1
cg
|D|n2 e(nτ ). d2
The Shimura correspondence has closely related to periods of cusp forms in the following:
On the Kohnen-Zagier formula in the case of level 4p m
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Theorem 4 (Kohnen [3]:Theorem 2). For any f (z) ∈ S2k (N ), the adjoint map of Sk,N,D with respect to the Petersson inner product is represented by
D ∗ f |Sk,N,D (z) = (−1)[k/2] 2k µ(t) t m≥1 (−1)k m≡0,1 (mod 4)
t|N
×t k−1 rk,Nt (f ; D, (−1)k mt 2 ) e(mz) where µ(t) is the Moebius function. Remark 2. The Shimura correspondence Sk,N,D preserves old forms and newforms and commutes with all Hecke operators. 5. The Kohnen-Zagier formula in the case level 4pm In this section, we extend the Kohnen-Zagier formula in the case of N = p m with an odd prime number p. In the first place, Kohnen-Zagier [5] and Kohnen [3] proved the following result. Theorem 5 (m = 0: Kohnen-Zagier [5], m = 1: Kohnen [3]). Let k be a positive k integer, m = 0, 1 and D be a fundamental discriminant such that (−1) D > 0 and (p, D) = 1. Suppose that f (z) = n≥1 af (n)e(nz) is a primitive form in new (p m ) such that f |W (p m ) = D f and S2k pm
g(τ ) =
cg (n)e(nτ )
n≥1 (−1)k n≡0,1 (mod 4) K is a Hecke eigen form in Sk+1/2 (p m )new , which has the same Hecke eigen values as f . Then we have
|cg (|D|)|2 (k − 1)! L(f, D, k) = 2m . |D|k−1/2 k g, g π f, f Secondly we consider the case of m ≥ 2. Hereafter we assume that k ≥ 2, m ≥ 2 and D a fundamental discriminant with (−1)k D > 0 and (D, p) = 1. 5.1. CASE I (m = 2n with a natural number n ≥ 2) K,(−1)i m new K m new In this case, Sk+1/2 (p ) = has the ‘multiplicity two Sk+1/2 (p ) i=0,1
condition’ by Theorem 2. Therefore we take two nontrivial Hecke eigen forms
K,+1 g+ (τ ) = cg+ (n)e(nτ ) ∈ Sk+1/2 (p m )new n≥1 (−1)k n≡0,1 (mod 4)
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and
g− (τ ) =
K,−1 ∈ Sk+1/2 (p m )new
cg− (n)e(nτ )
n≥1 (−1)k n≡0,1 (mod 4)
so that g+ , g+ = g− , g− , which have the same Hecke eigen values as a primitive form
new,+1 m af (n)e(nz) ∈ S2k (p ). f (z) = n≥1
Put g(τ ) = g+ (τ ) + g− (τ ) and g (τ ) = g|Rp (τ ) = g+ (τ ) − g− (τ ). Then we see that g = ±g because of the method of taking them and g, g are Hecke eigen forms too. Furthermore we put h1 (τ ) = g(τ ),
h2 (τ ) = g(τ ) −
g, g g (τ ). g , g
These forms h1 , h2 are Hecke eigen orthogonal bases on the space K,±1 K m m new Sk+1/2 (p , f ) = g(τ ) ∈ Sk+1/2 (p ) g|Tk+1/2,4pm (l 2 )(τ ) = af (l)g(τ ) for prime l with (l, p) = 1 , K,±1 where Tk+1/2,4pm (l 2 ) are l 2 -th Hecke operators on Sk+1/2 (p m )new . From definitions of f and hi , we have
hi |Sk,pm ,D (z) = chi (|D|)f (z)
(i = 1, 2),
and hence Fourier coefficients of f and hi are related as
D 2 chi (n |D|) = chi (|D|) µ(d) d k−1 af (n/d). d d|n,(d,N)=1
Therefore we have the following chi (|D|)hi |Sk,pm ,D (z) = |chi (|D|)|2 f (z)
(i = 1, 2),
that is, 2 2
chi (|D|) |chi (|D|)|2 hi |Sk,pm ,D (z) = f (z). hi , hi hi , hi i=1
i=1
Taking the Petersson inner product of each side and f , we obtain the following 2 2
chi (|D|) |chi (|D|)|2 f, hi |Sk,pm ,D = f, f . hi , hi hi , hi i=1
i=1
On the Kohnen-Zagier formula in the case of level 4p m
263
∗ But the left hand side of it equals to the |D|-th Fourier coefficient of f |Sk,N,D K because hi (i = 1, 2) construct a normal orthogonal system of Sk+1/2 (p m , f ). Therefore we get 2
|chi (|D|)|2 f, f = (−1)[k/2] 2k rk,N (f ; D, D) hi , hi i=1
by Theorem 4. Secondly, we calculate rk,N (f ; D, D) by the same manner in Kohnen-Zagier [5] and Kohnen [3]. A representative system of Qpm ,D 2 are given by {[0, D, µ], [0, D, µ] ◦ W (pm )|µ mod D} because of ‘reducing method’ (cf. Gross-Kohnen-Zagier [1], Chap. 1). Therefore, noting that f |W (p m )(z) = f (z) and D m ωD ([0, D, µ]) = ωD ([0, D, µ]), ωD ([0, D, µ] ◦ W (p )) = pm we get the following equality
ωD ([0, D, µ] ◦ Wt ) rk,N (f ; D, D) = t=1,pm µ mod D
×
f (z)([0, D, µ] ◦ Wt (z, 1))k−1 dz γ[0,D,µ]◦Wt
i∞ D f (z)(Dz + µ)k−1 dz µ −µ/D µ mod D ∞ D D 1/2 k−1 1/2 |D| af (n)e−2πnt t k−1 dt = 2(Di) i −1 n 0 =2
n≥1
= 2(−1)
[k/2]
(2π)
−k
|D|
k−1/2
Γ (k)L(f, D, k)
where L(f, D, s) is the twisted L-function
D af (n)n−s L(f, D, s) = n
(Re(s) >> 0).
n≥1
On the other hand, we see the following 2 g, g − g|Rp , g |D|
p |chi (|D|)|2 |cg (|D|)|2 =2 2 hi , hi g, g − g|Rp , g2 i=1
by the straightforward way. Finally we have the result of Kohnen-Zagier type in the case of N = pm . Theorem 6. Let the notation and assumption be as above. For any primitive form
new,+1 m af (n)e(nz) ∈ S2k (p ), f (z) = n≥1
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we can take g(τ ) =
K cg (n)e(nτ ) ∈ Sk+1/2 (p m , f )
n≥1 (−1)k n≡0,1 (mod 4)
so that g|Rp = ±g and g, g = ±g|Rp , g, which has the same Hecke eigen values as f . Furthermore we have the relation g, g − g|Rp , g |D| p L(f, D, k) (k − 1)! |cg (|D|)|2 = |D|k−1/2 g, g2 − g|Rp , g2 πk f, f for them. Remark 3. Theorem 6 has the same form as Theorem 5. Because we have the following: 2 |cg+ (|D|)| if |D| = 1 g, g − g|Rp , g |D| p 2g ,g + p |cg (|D|)|2 = |c +(|D|)| 2 2 2 g |D| − g, g − g|Rp , g = −1, 2g− ,g− if p in the straightforward way. In other words, by changing D, we get the central value L(f, D, k) in either one coefficients of the two different forms alternatively. 5.2. CASE II (m = 2n + 1 with a natural number n)
new,(−1)i
−1 p
We take two primitive forms fi (z) = n≥1 afi (n)e(nz) ∈ S2k (i = 0, 1). Then, we can take two nontrivial Hecke eigen newforms
K,(−1)i cgi (n)e(nτ ) ∈ Sk+1/2 (p m )new , gi (τ ) =
k
(p m )
n≥1 (−1)k n≡0,1 (mod 4)
which correspond to fi by the Shimura correspondence respectively, because K Sk+1/2 (p m )new in this case has multiplicity one theorem by Theorem 2. Therefore we have the relation |cgi (|D|)|2 L(fi , D, k) (k − 1)! i |D| |D|k−1/2 = 1 + (−1) (i = 0, 1) k gi , gi p π fi , fi in the same argument as Theorem 5,6. Hence Theorem 7. The notation being as above, we take an arbitrary Hecke eigen form
cg (n)e(nτ ) g(τ ) = n≥1 (−1)k n≡0,1 (mod 4)
On the Kohnen-Zagier formula in the case of level 4p m K in Sk+1/2 (p m )new . Put gi =
1 2
265
g + (−1)i g|Rp for i = 0, 1. Then we have
1 1
|cgi (|D|)|2 L(fi , D, k) (k − 1)! i |D| 1 + (−1) |D|k−1/2 = , k gi , gi p π fi , fi i=0
i=0
new,(−1)i
where fi is the primitive form in S2k eigen values as gi .
−1 p
k
(p m ), which has the same Hecke
Remark 4. The above relation shows that Kohnen-Zagier’s formula in this case has the same form as multiplicity one case. 5.3. CASE III (m = 2) Firstly, we prepare for the following lemma. Lemma 1. For any f ∈ S2k (1) ⊕ S2k (p)new , we have −1 2 2 f |Rp ∈ S2k (p ) and f |Rp W (p ) = f |Rp . p We can prove this lemma by straightforward calculation and the theory of newforms (of integral weight). Nextly, we take a primitive form
new 2 f (z) = af (n)e(nz) ∈ S2k (p ). n≥1
Then we have −1 t (f ) f, f |W (p2 ) = p
where t (f ) =
new (p)|R 1 iff ∈ S2k (1)|Rp ⊕ S2k p ∗,+ 2 2 iff ∈ S2k (p )
by the above lemma. On the other hand, we can take two nontrivial Hecke eigen K,+1 K,−1 forms g+ (τ ) ∈ Sk+1/2 (p 2 )new and g− (τ ) ∈ Sk+1/2 (p 2 )new so that g+ , g+ = g− , g− , which have the same Hecke eigen values as f by using Theorem 3. Therefore we have the similar formula to Theorem 6 in the same argument as it, i.e., Theorem 8. Let the notation be as above. Put g = g+ + g− . Then we arrive at the conclusion t (f ) g, g − g|Rp , g |D| p −1 1 1+ |cg (|D|)|2 = g, g2 − g|Rp , g2 2 p ×
(k − 1)! L(f, D, k) . |D|k−1/2 k π f, f
Acknowledgements. The author would like to thank Professor H. Kojima and the referee for pointing out some unsuitable expressions in the earlier versions, and to Professor U. Jannsen for his meticulous and conscientious proof-reading and checking of my article.
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