Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2018: 0.34

ICV 2018: 152.31

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 15, Issue 1

Issues

Volume 13 (2015)

Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications

SoYoung Choi
  • Department of Mathematics Education and RINS, Gyeongsang National University, 501 Jinjudae-ro, Jinju, 660-701, South Korea
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Chang Heon Kim
Published Online: 2017-03-30 | DOI: https://doi.org/10.1515/math-2017-0020

Abstract

For an odd and squarefree level N, Kohnen proved that there is a canonically defined subspace Sκ+12new(N)Sκ+12(N),andSκ+12new(N)andS2knew(N) are isomorphic as modules over the Hecke algebra. Later he gave a formula for the product ag(m)ag(n)¯ of two arbitrary Fourier coefficients of a Hecke eigenform g of halfintegral weight and of level 4N in terms of certain cycle integrals of the corresponding form f of integral weight. To this end he first constructed Shimura and Shintani lifts, and then combining these lifts with the multiplicity one theorem he deduced the formula in [2, Theorem 3]. In this paper we will prove that there is a Hecke equivariant isomorphism between the spaces S2k+(p)andSk+12(p). We will also construct Shintani and Shimura lifts for these spaces, and prove a result analogous to [2, Theorem 3].

Keywords: Modular forms; Shintani lifts; Shimura lifts

MSC 2010: 11F11; 11F67; 11F37

1 Introduction and statement of results

For a positive integer N, we let Γ0(N):={abcdSL2(Z)|c0(modN)}

be the Hecke subgroup of Γ(1):=SL2(ℤ), and Γ0+(N) be the group generated by the Hecke group Γ0(N) and the Fricke involution WN:=01/NN0. If k is a positive integer, then we write S2k(N)(resp. S2k+(N)) for the space of cusp forms of weight 2k on Γ0(N)(resp. Γ0+(N)). We also denote by sk+12(N) the space of cusp forms of weight k+12 on Γ0(4N), which have a Fourier expansion Σn≥1 a(n)qn with a(n) = 0 unless (–1)kn ≡ 0, 1 (mod 4). Let Sk+12(N) be the subspace of Sk+12(N), in which the Fourier expansion of each form is supported only on those n ∈ ℤ for which (–1)kn ≡ □ (mod 4N). For each prime divisor p of N we define sk+12±,p(N) as the subspace of Sk+12(N) consisting of forms whose n-th Fourier coefficients vanish for (1)knp=1. In [1, Proposition 4] Kohnen defines, for each prime divisor p of N, a Hermitian involution wp,k+12NonSk+12(N) and shows that its (±)-eigenspace is equal to Sk+12±,p(N). Thus we have an orthogonal decomposition sκ+12(N)=sκ+12+,p(N)sκ+12,p(N).

We then observe that Sk+12+,p(p)=Sk+12(p).

Let S2knew(N) be the subspace of newforms in S2k(N). We assume that the level N is odd and squarefree. In [1, page 65] Kohnen proved that there is a canonically defined subspace Sk+12new(N)Sk+12(N),andSk+12new(N) and S2knew(N) are isomorphic as modules over the Hecke algebra. Later in [2, Theorem 3] he gave a formula for the product ag(m)ag(n)¯ of two arbitrary Fourier coefficients of a Hecke eigenform g of half-integral weight and of level 4N in terms of certain cycle integrals of the corresponding form f of integral weight. To this end he first constructed Shimura and Shintani lifts, and then combining these lifts with the multiplicity one theorem [1, Theoerm 2] he deduced the formula in [2, Theorem 3].

In this paper we will prove in Theorem 1.10 that there is a Hecke equivariant isomorphism between the spaces S2k+(p)andSk+12(p). We will also construct Shintani and Shimura lifts for these spaces (see Theorems 1.6 and 1.9), and prove in Theorem 1.11 a result analogous to [2, Theorem 3].

In the following, we introduce Poincaré series, Petersson scalar products, Shintani and Shimura lifts, and develop their properties relavant to our settings. Let ℌ denote the complex upper half plane. If f and g are cusp forms of weight κ12Z on some subgroup Γ of finite index in Γ(1) we denote their Petersson product by f,g=f,gΓ=1[Γ(1):Γ]ΓHf(z)g(z)¯yκ2dxdy(x=Rez,y=Imz).

Let m be a positive integer such that (— 1)km = □ (mod 4p). Following [3], for z ∈ ℌ, we let Pm+(z)Sk+12(p) be a Poincaré series such that g,Pm+(z)=ag(m)c(m)for allg=n1ag(n)qnSk+12(p),

where c(m)=Γ(k12)i4p1(4πm)k12. Here iN stands for [Γ(1) : Γ0(N)]. Let α be a positive integer with (–1)k α ≡ 0, 1 (mod 4). As described in [2, (4)], for z ∈ ℌ, let Pk,p,α(z)Sk+12(p) be the Poincaré series characterized by g,Pk,p,α(z)=ag(α)c(α)for allg=n1ag(n)qnSk+12(p).

Write Pk,p,α=Pk,p,α+Pk,p,αSk+12(p)Sk+12,p(p).

Proposition 1.1

Let α be a positive integer with (–1)k α ≡ 0, 1 (mod 4).

(i) If (–1)k α ≡ □ (mod 4p), then Pk,p,α+=Pα+.

(ii) (ii) If (–1)k α ≢ D (mod 4p), then Pk,p,α+ is identically zero.

(iii) If (–1)k α ≡ □ (mod 4p) with p α then Pk,p,α=Pα+.

The motivation of this paper is as follows. In [3] we have shown that the space Sk+12(p) is spanned by the Poincaré series Pα+. In [1, 2] Kohnen constructed Shimura lift by making use of the Poincaré series Pk,p,α in the space Sk+12(p). So we expect that these Poincaré series Pα+ can be used to find certain space of cusp forms of integral weight corresponding to the space Sk+12(p) under Shimura and Shintani lifts.

Let 𝔇 be the set of all discriminants, i.e. D={dZ|d0,1(mod4)}.

For d ∈ 𝔇, we let 𝓠d,N be the set of all integral binary quadratic forms Q(x,y) = ax2 + bxy + cy2 = [a, b, c] with N | a and b2 — 4ac = d. We observe that 𝓠d,N is an empty set unless d is congruent to a square modulo 4N. For M=αβγδΓ0+(N), we set X′ = αX + βY, Y′ = γ X+ δY, and Q(X, Y)∘ M = Q(X′, Y′). This defines an action of the group Γ0+(N) on the set 𝓠d,N. Following [2, page 239], for integers k ≥ 2, N ≥ 1 and D, D′ ∈ 𝔇 with DD′ > 0 we set fk,N(z;D,D):=QQDD,NχD(Q)Q(z,1)k(zH),

where 𝓧d(Q)is the generalized genus character defined as follows: for Q = [a, b, c], χD(Q):=0if(a,b,c,D)>1Drif(a,b,c,D)=1,where[a,b,c]representsr,(r,D)=1.

The series converges absolutely uniformly on compact sets, and fk,N(z; D, D′) is a cusp form of weight 2k on Γ0(N).

Remark 1.2

(i) The series fk,N(z; D, D’) is a cusp form of weight 2k on Γ0+(N) since the character χD is invariant under the Fricke involution WN (see [4, Proposition 1 in p.508]).

(ii) The series fk,N(z; D, D′) is identically zero unless DD′ ≡ □ .(mod 4N) and (–1)kD > 0.

Proposition 1.3

For cusp forms f and g of weight κ12ZonΓ0+(p), we have Γ0(p)Hfg¯yκ2dxdy=2Γ0+(p)Hfg¯yκ2dxdy.

If d ∈ 𝔇 is positive and is not a perfect square, then the group Γ0+(p)Q/{±1}={gΓ0+(p)|Qg=Q}/{±1}

is infinite cyclic with a distinguished generator gQ which is explicitly described in [5, Theorem 1.3]. For Q = [a, b, c]∈ 𝓠d.p let SQ be the oriented semi-circle defined by a|z|2 + (Re z) b + c = 0, directed counterclockwise if a > 0 and clockwise if a < 0. For fS2k+(p) and ∈ 𝓠d,p we define rQ+(f):=cQ+f(z)dQ,kz

where cQ+=cQ+(z0) is the directed arc on SQ from z0SQ to gQz0, and dQ,kz = Q(z, 1)k1 dz. Then the value rQ+(f) is both independent of z0SQ and a class invariant.

For each fS2k+(p) and integers D, m satisfying the following condition (1)kmD,(1)kD>0,m>0,|D|m,and|D|m(mod4p),(1)

we set rk,p+(f;D,(1)km):=QQ|D|m,p/Γ0+(p)χD(Q)rQ+(f).

We also define for cusp forms f and g of weight 2k ∈ 2ℤ on Γ0+(p), f,gΓ0+(p)=2ipΓ0+(p)Hfg¯y2k2dxdyso thatf,gΓ0+(p)=f,gΓ0(p).

Proposition 1.4

For any fS2k+(p) and integers D, m satisfying (1) we have f,fk,p(;D,(1)km)=f,fk,p(;D,(1)km)Γ0+(p)=2ipπ2k2k122k+2(|D|m)12krk,p+(f;D,(1)km).

Proposition 1.5

For D ∈ 𝔇 with (–1)k D > 0, we let φk,p(z,τ):=m1(1)km0,1(4)mk12fk,p(z;D,(1)km)e2πimτ.

Then for fixed z, φk,p(z, τ) is an element of Sk+12(p).

Let wp:=wp,k+12p be the Hermitian involution (with respect to τ) on the space Sk+12(p). We then have a decomposition φk,p(z,τ)=φk,p+(z,τ)φk,p(z,τ)Sk+12(p)Sk+12,p(p).

Theorem 1.6

Let D ∈ 𝔇 with (1)k D > 0 and D ≡ □ (mod 4p). Define for each fS2k+(p),, the D-th Shintani lift f|S+:=ipck,Df,φk,p+(,τ¯)Sk+12(p),

with ck,D=ip(1)[k2]3(2π)k(k1)!c(|D|). If we write the Fourier expansion of g:=f|S+ as g(τ)=m1(1)km(mod4p)ag(m)e2πimτ,

then for a positive integer m satisfying p m and (–1)km ≡ □ .(mod 4p) one has ag(m)=2rk,p+(f;D,(1)km).

Remark 1.7

(i) In the case p = 1 and k = 0, cycle integrals of the modular invariant j -function were considered in [6] and their generating function was shown to be a mock modular form of weight 1/2 on Γ0(4).

(ii) In [5, 7] cycle integrals of weakly holomorphic modular forms were used to construct modular integrals for certain rational period functions related to indefinite binary quadratic forms.

We have φk,p+(z,τ)=φk,p(z,τ)+φk,p(z,τ)|wp2.

Alternatively, we can find φk,p+(z,τ) by means of Poincaré series as follows. Following [2, (3)], for z ∈ ℌ and D ∈ 𝔇 with (–1)k D > 0, we let Ωk,p(z,τ;D)=ipck,D1m1(1)km0,1(4)mk12t|pμ(t)Dttk1fk,pt(tz;D,(1)km)e2πimτ

where ck,D=(1)[k2]|D|k+12π2k2k123k+2 and μ(t) is the Möbius μ-function. By [2, (3)] or (5) one has φk,p(z,τ)=ip1ck,DΩk,p(z,τ;D)+pk1ipDpΩk,1(pz,τ;D)=(1)[k2]3(2π)k(k1)!n1nk1d|npdDdndkPk,p,n2|D|/d2(τ)e2πinz+pk1(Dp)(1)[k2]3(2π)k(k1)!n1nk1d|nDdndkPk,1,n2|D|/d2(τ)e2πinpz.

This gives rise to the following theorem.

Theorem 1.8

Let D ∈ 𝔇 with (–1)kD > 0 and D ≡ □ (mod 4p). We have φk,p+(z,τ)=(1)[k2]3(2π)k(k1)!n1nk1d|npdDdndkPk,p,n2|D|/d2+(τ)e2πinz+pk1Dp(1)[k2]3(2π)k(k1)!n1nk1d|nDdndkPk,1,n2|D|/d2+(τ)e2πinpz.(2)

Let D ∈𝔇with (–1)k D > 0 and D ≡ □ (mod 4p). Now we define for each gSk+12(p), the D-th Shimura lift g|S+:=ipck,Dg,φk,p+(z¯,τ)τS2k+(p).

It then follows from (2) that g|S+=ipck,D(1)[k2]3(2π)k(k1)!n1nk1d|npdDd(nd)kag(n2|D|/d2)c(n2|D|/d2)e2πinz+pk1ipck,DDp(1)[k2]3(2π)k(k1)!n1nk1d|nDdndkg,Pk,1,n2|D|/d2+(τ)τe2πinpz.

Theorem 1.9

(i) Our Shimura lift is Hecke equivariant, i.e. for each gSk+12(p) and a prime r(≠ p), we have (g|S+)|T(r)=(g|T(r2))|S+

(ii) Our Shintani lift is Hecke equivariant, i.e. for each fS2k+(p) and a prime r(≠ p), we have (f|S+)|T(r2)=(f|T(r))|S+.

Theorem 1.10

Let p be an odd prime. There exists a Hecke equivariant isomorphism L´(=L´p):S2k+(p)Sk+12(p).

Theorem 1.11

Let D ∈ 𝔇) with (1)k D > 0 and D ≡ □ (mod 4p). Let p be an odd prime and L´(=L´p):S2k+(p)Sk+12(p). be the Hecke equivariant isomorphism discussed in Theorem 1.10 and let {f1, . . . , ft} be a basis for S2k+(p) consisting of normalized Hecke eigenforms. For each j ∈ {1, . . . ,t} we put gj := Ĺ (fj) Then the following assertions are true.

(i) Let 𝓢+ denote the D -th Shimura lift. Then gj|S+=cj(|D|)fj

where cj.(|D|) denotes the Fourier coefficient of q|D| in gj.

(ii) For each positive integer n, the Fourier coefficients cj (n) of qn in gj satisfy cj(n)¯=ccj(n)

for some nonzero constant c.

(iii) Let S+ denote the D -th Shintani lift. Then fj|S+=cj(|D|)¯fj,fjgj,gj1gj.

(iv) For each positive integer m such that (–1)km ≡ □ (mod 4p) and p m, we have rk,p+(fj,D,(1)km)=12cj(|D|)¯cj(m)fj,fjgj,gj1

(v) For each fS2k+(p)andgSk+12(p), one has g|S+,f=g,f|S+,

i.e. 𝓢+ and S+ are adjoint.

Remark 1.12

(i) Let Sk,p,D denote the Kohnen’s Shintani lift [2, (8)] and f be a newform in S2k+(p). Then our Shintani lift f|S+ is the projection of Kohnen’s Shintani lift f|Sk,p,DonSk+12(p) up to multiplication. But they are different for oldforms.

(ii) Let 𝓢k,p,D denote the Kohnen’s Shimura lift [2, (6)] and g be a newform in Sk+12(p) Our Shimura lift g |𝓢+ and Kohnen’s Shimura lift g |𝓢k,p,D are the same up to multiplication. But they are different for oldforms.

(iii) Kohnen’s results [1] do not give a Hecke equivariant isomorphism between the spaces S2k+(p)andSk+12(p) while we give it in Theorem 1.10.

(iv) The results in Theorem 1.11 hold for all Hecke eigenforms in S2k+(p)orSk+12(p) But when we consider the spaces S2k+(p)andSk+12(p) Kohnen’s results hold only for newforms.

Throughout this paper, given a modular form f we denote by af (n), the Fourier coefficient of qn in f. This paper is organized as follows. The proofs of Propositions 1.1, 1.3, 1.4 and 1.5 are given in Section 2. In Section 3 the proofs of Theorems 1.6 and 1.9 are given. In Section 4 and Section 5 we prove Theorems 1.10 and 1.11, respectively.

2 Proofs of Propositions 1.1,1.3,1.4 and 1.5

Proof of Proposition 1.1. We observe that for fSk+12(p)andghSk+12(p)=Sk+12(p)sk+12,p(p), f,g+h=f,g.

Write Pk,p,α+(τ)=(1)kl(4p)aPk,p,α+(l)e2πilτ. We then have for each positive integer l with (–1)kl ≡ □ (mod 4p), aPl+(α)c(α)=Pl+,Pk,p,α=Pl+,Pk,p,α+=Pk,p,α,+Pl+¯=c(l)aPk,p,α+(l)¯,

so that aPk,p,α+(l)=c(α)c(l)aPl+(α)¯,(3)

which is zero unless (–1)k α ≡ □ (mod 4p). This proves the second assertion. To prove the first assertion we observe that aPl+(α)c(α)=Pl+,Pα+=Pα+,Pl+¯=aPα+(l)¯c(l).(4)

Combining (3) and (4) we get that aPk,p,α+(l)=aPα+(l),, from which the first assertion is immediate. Now we prove the third assertion. For each positive integer α and β with (–1)k α ≡ □ (mod 4p)and (–1)k β ≡ □ (mod 4p), we consider Pk,p,α,Pk,p,β=Pk,p,αPα+,Pk,p,βPβ+=c(β)aPk,p,α(β)c(α)aPβ+(α)¯=c(β)aPk,p,α(β)c(β)aPα+(β)by (4)=c(β)aPk,p,α(β).

Thus Pk,p,α,Pk,p,β=Pα++Pk,p,α,Pk,p,β=Pk,p,α,Pk,p,β =c(β)aPk,p,α(β)=c(α)aPk,p,β(α)¯=0sincepαand(1)kαp=1.

This implies that aPk,p,α(β)=0 for arbitrary positive integer β with (–1)k β ≡ □ (mod 4p)so that Pk,p,α=Pα+.

Proof of Proposition 1.3. Let Fp+ be a fundamental domain of Γ0+(p). Then Fp+WpFp+ is a fundamental domain of Γ0(p). Thus we see that Γ0(p)Hfg¯yκ2dxdy=Fp+fg¯yκ2dxdy+WpFp+fg¯yκ2dxdy=Fp+fg¯yκ2dxdy+Fp+f|κWpg|κWp¯yκ2dxdy=2Fp+fg¯yκ2dxdy=2Γ0+(p)Hfg¯yκ2dxdy,

as desired.

Proof of Proposition 1.4. For a Γ0+(p)-class 𝔘 of forms Q ∈ 𝓠|D|m,p, we define fk,p(z;U)=QUQ(z,1)k.

We then have fk,p(z;U)S2k+(p) and ipf,fk,p(;U)Γ0+(p)=2Γ0+(p)HγΓQΓ0+(p)(Qγ)(z¯,1)kf(z)y2k2dxdy=2ΓQHQ(z¯,1)kf(z)y2k2dxdy,

where Q is any element of 𝔘 and ΓQ is the stabilizer of Q in Γ0+(p). By similar argument in [2, Proposition 7], we obtain f,fk,p(;U)Γ0+(p)=2ip(|D|m)12k0πsin2k2θdθcQ+f(z)dQ,kZ

and hence f,fk,p(;D,(1)km)Γ0+(p)=2ipπ2k2k122k+2(|D|m)12krk,p+(f;D,(1)km).

Proof of Proposition 1.5. We observe that Ωk,p(z,τ;D)=ipck,D1m1(1)km0,1(4)mk12fk,p(z;D,(1)km)e2πimτipck,D1m1(1)km0,1(4)mk12Dppk1fk,1(pz;D,(1)km)e2πimτSk+12(p).

Since Ωk,1(z,τ;D)=ck,D1m1(1)km0,1(4)mk12fk,1(z;D,(1)km)e2πimτ

belongs to Sk+12(1)Sk+12(p)for fixed τ, one has Ωk,1(pz,τ;D)=ck,D1m1(1)km0,1(4)mk12fk,1(pz;D,(1)km)e2πimτSk+12(p).

Thus we obtain that φk,p(z,τ)=m1(1)km0,1(4)mk12fk,p(z;D,(1)km)e2πimτ=ip1ck,DΩk,p(z,τ;D)+pk1ipDpΩk,1(pz,τ;D)Sk+12(p),(5)

as desired.

3 Proofs of Theorems 1.6 and 1.9

Proof of Theorem 1.6. We note that for each positive integer m such that (–1)km ≡ □ (mod 4p)and p m, aφk,p+(z,τ)(m)=mk12fk,p(z;D,(1)km).

Thus we have g(τ)f|S+=ipck,Df,φk,p+(,τ¯)=ipck,Df,m1,pm(1)km(4p)mk12fk,p(;D,(1)km)e2πimτ¯+m1,p|m(1)km(4p)ag(m)e2πimτ=ipck,Dm1,pm(1)km(4p)mk12f,fk,p(;D,(1)km)e2πimτ+m1,p|m(1)km(4p)ag(m)e2πimτ=2m1,pm(1)km(4p)rk,p+(f;D,(1)km)e2πimτ+m1,p|m(1)km(4p)ag(m)e2πimτ by Proposition 1.4

from which the assertion is immediate.

Proof of Theorem 1.9. First we need a lemma.

Lemma 3.1

One has (φk,p|T(r2))+=φk,p+|T(r2)(6)

and (φk,p|T(r))+=φk,p+|T(r).(7)

Proof

We observe that (φk,p=φk,p+φk,p)|T(r2)=φk,p+|T(r2)φk,p|T(r2)Sk+12(p)Sk+12,p(p),

from which (6) is immediate. By a similar reasoning one obtains (7).

For the proof of the first assertion of Theorem 1.9 we observe that for l ∈ {1, p} and each gSk+12(l), g,Ωk,l(z¯,,D)|T(r2)={g|T(r2),Ωk,l(z¯,,D)}=(g,Ωk,l(z¯,,D))|T(r)by[2,p.241]=g,Ωk,l(z¯,,D)|T(r).

Thus one obtains Ωk,l(z¯,τ,D)|T(r2)=Ωk,l(z¯,τ,D)|T(r)(8)

where the operator T(r2)(resp. T(r))acts on a function with variable τ(resp. z). In particular, Ωk,1(pz¯,τ,D)|T(r2)=Ωk,1(z¯,τ,D)|T(r2)|V(p)=Ωk,1(z¯,τ,D)|T(r)|V(p)=Ωk,1(z¯,τ,D)|V(p)|T(r)=Ωk,1(pz¯,τ,D)|T(r)(9)

where the operator V(p) acts on a function with variable z. Consequently, for gSk+12(p), (g|S+)|T(r)=(g,φk,p+(z¯,τ)τ)|T(r)=(g,φk,p(z¯,τ)τ)|T(r)=g,ip1ck,DΩk,p(z¯,τ,D)|T(r)+g,pk1ck,DDpΩk,1(pz¯,τ,D)|T(r)by(5)=g,ip1ck,DΩk,p(z¯,τ,D)|T(r)+g,pk1ck,DDpΩk,1(pz¯,τ,D)|T(r)=g,ip1ck,DΩk,p(z¯,τ,D)|T(r2)+g,pk1ck,DDpΩk,1(pz¯,τ,D)|T(r2)by(8)and(9)=g|T(r2),φk,p(z¯,τ)=g|T(r2),φk,p+(z¯,τ)=(g|T(r2))|S+

Next we will prove the second assertion of Theorem 1.9. For fS2k(l), let f |S*denote the D-th Shintani lifting of f, defined in [2, p.240]. We then remark from [2, Theorem 2] that f,Ωk,l(,τ¯,D)=f|SforallfS2k(l)(10)

and (f|T(r))|S=(f|S)|T(r2)forallprimer(l).(11)

Thus for all f ∈ S2k(l), we have that f,Ωk,l(,τ¯,D)|T(r)=f|T(r),Ωk,l(,τ¯,D)=f,Ωk,l(,τ¯,D)|T(r2)by(10)and(11)=f,Ωk,l(,τ¯,D)|T(r2),

which implies that Ωk,l(z,τ¯,D)|T(r)=Ωk,l(z,τ¯,D)|T(r2).

In particular, Ωk,1(pz,τ¯,D)|T(r)=Ωk,1(z,τ¯,D)|V(p)|T(r)=Ωk,1(z,τ¯,D)|T(r)|V(p)=Ωk,1(pz,τ¯,D)|T(r2).

Hence we obtain that for all fS2k+(p), ck,Dip(f|S+)|T(r2)=f,φk,p+(,τ¯)|T(r2)=f,φk,p+(,τ¯)|T(r2)=f,(φk,p(,τ¯)|T(r2))+by(6)=f,(φk,p(,τ¯)|T(r))+=f,φk,p+(,τ¯)|T(r)by(7)=f|T(r),φk,p+(,τ¯)=ck,Dip(f|T(r))|S+.

4 Proof of Theorem 1.10

Following [1, p.64 lines 7-9], let Sk+12old(p) be the set of oldforms in Sk+12(p). Let Sk+12old(p):=Sk+12old(p)Sk+12(p).

Lemma 4.1

Define a map Φ:Sk+12(1)Sk+12old(p)

by Ф (f) = f + f |wp. Then Ф is a Hecke equivariant isomorphism.

Proof

According to [1, p.65 line 4], the Hermitian involution wp leaves the space of oldforms stable. Let fSk+12(1). It then follows from [1, p.66 line 5] that f | wp is an oldform, too. Therefore we have an inclusion {f+f|wp|fSk+12(1)}Sk+12o1d(p)Sk+12(p)=Sk+12old(p).

While [8, Theorem 5.11 in p.2612] says Sk+12(p)=Sk+12new(p){f+f|wp|fSk+12(1)},

in [1] it is shown that Sk+12(p)=Sk+12new(p)Sk+12old(p).

Thus we have Sk+12old(p)={f+f|wp|fSk+12(1)}.(12)

It follows from (12) that the map Ф is surjective. Next we will show that Φ is injective. Indeed if we assume Ф (f) = f + f | wp = 0 for f=n1a(n)qnSk+12(1), then we have 0=f+f|wp=n1a(n)qn+n1(1)knpa(n)+pka(n/p2)qnby[1,(44)].

Comparing the coefficients of qp2m we obtain that a(p2m) = –pk a(m)for all positive integers m, which means that f | U (p2) = –pk f. But it follows from [1, Theorem 2] that Sk+12(p)=Sk+12(1)Sk+12(1)|U(p2)Sk+12new(p),

from which we deduce that f = 0. This shows that Ф is injective. Finally we will show that Ф is Hecke equivariant. By [1, lines 2-3 in p.66] we have f|wp=p1k(f|U(p2)+f|Tk+12,1(p2).

Thus for any prime r(≠ p), one has Φ(f)|T(r2)=(f+f|wp)|T(r2)=f|T(r2)+(f|wp)|T(r2)=f|T(r2)+(f|T(r2))|wpsinceU(p2)andT(p2)commutewithT(r2)=Φ(f|T(r2).

Let Φ1:S2k(1)Sk+12(1) be the Kohnen-Shimura isomorphism which is Hecke equivariant, and we let S2k(1)S2k+old(p) be a Hecke equivariant isomorphism defined by Ф2(f) = f + f |Wp. These facts combined with Lemma 4.1 show that the map φ:=ΦΦ1Φ21 defines a Hecke equivariant isomorphism from S2k+old(p)toSk+12+old(p). Meanwhile, according to [1, page 71] there exists a Hecke equivariant isomorphism from S2k+new(p)toSk+12new(p). Now we see that the map L´:S2k+(p)(=S2k+old(p)S2k+new(p))Sk+12(p)

defined by L´(fg)=φ(f)ψ(g)

is a Hecke equivariant isomorphism.

5 Proof of Theorem 1.11

(i) Utilizing Theorems 1.9 and 1.10 we get that for each prime r(∘ p), (gj|S+)|T(r)=(gj|T(r2))|S+=L´(fj|T(r))|S+=λj(r)L´(fj)|S+=λj(r)gj|S+,

for some λj (r)∈ ℂ By multiplicity one theorem we have gj|S+=cfjfor somecC.(13)

Comparing Fourier cofficient of q on both sides of (13) we obtain c = cj (| D |), which proves the assertion.

(ii) We note that for each prime r(≠ p), L´1(gj|T(r2))=L´1(gj)|T(r)=fj|T(r)=λj(r)fj=L´1(λj(r)gj),

so that gj|T(r2)=λj(r)gj(14)

Thus we have gj(τ¯)¯|T(r2)=(gj|T(r2))(τ¯)¯=λj(r)gj(τ¯)¯

and therefore L´1(gj(τ¯)¯)|T(r)=L´1(gj(τ¯)¯|T(r2))=λj(r)L´1(gj(τ¯)¯).

This implies that L´1(gj(τ¯)¯)=cfj=cL´1(gj)for some nonzerocC.

Hence we have gj(τ¯)¯=cgj(τ)

and therefore cj(n)¯=ccj(n).

(iii) We note that g1, . . . , gt are mutually orthogonal with respect to Petersson scalar product. Write φk,p+(z,τ)=j=1thj(z)gj(τ).

We then have φk,p+(z¯,τ)=j=1thj(z¯)gj(τ),

so that gj|S+=gj,φk,p+(z¯,τ)τ=hj(z¯)¯gj,gj.

Meanwhile by the assertion (i), one has gj| S+ = cj (| D|) fj. Thus we come up with hj(z)=cj(|D|)¯gj,gj¯fj(z¯)¯=cj(|D|)¯gj,gj¯fj(z),

since fj has real Fourier coefficients. Hence we get that φk,p+(z,τ)=j=1tcj(|D|)¯gj,gj¯fj(z)gj(τ),

and therefore ck,Dipfj|S+=fj(z),φk,p+(z,τ¯)z=cj(|D|)fj,fjgj,gjgj(τ¯)¯=cj(|D|)fj,fjgj,gjgjby the assertion (ii).

(iv) Recalling from Theorem 1.6 that Fj:=fj|S+=m1,pm(1)km(4p)2rk,p+(fj;D,(1)km)e2πimτ+m1,p|m(1)km(4p)aFj(m)e2πimτ,

we see that the assertion (iv) is immediate from (iii). (v) First we claim that for each i, j ∈ {1, . . . , t}, one has gi|S+,fj=gi,fj|S+.

Indeed we obtain that gi|S+,fj=cj(|D|)fi,fj=cj(|D|)δijfi,fj

and by the assertion (iii) gi,fj|S+=gi,gjcj(|D|)fj,fjgj,gj=cj(|D|)δijfi,fj.

More generally if we write g=i=1taigiandf=j=1tbjfj, then g|S+,f=i=1taigi|S+,j=1tbjfj=i,jajbj¯gi|S+,fj=i,jajbj¯gi,fj|S+by the claim=g,f|S+.

Acknowledgement

We appreciate the referees for their valuable comments. We also would like to thank KIAS (Korea Institute for Advanced Study) for its hospitality.

Choi was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014R1A1A2054915). Kim was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (NRF-2015R1D1A1A01057428 and 2016R1A5A1008055).

References

  • [1]

    W. Kohnen, Newforms of half-integral weight, J. reine angew. Math. 333 (1982), 32–72. Google Scholar

  • [2]

    W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), 237–268. CrossrefGoogle Scholar

  • [3]

    S. Choi and C. H. Kim, Linear relations among half-integral weight Poincaré series, J. Math. Anal. Appl. 432 (2015), 1077–1094. CrossrefGoogle Scholar

  • [4]

    B. Gross, W. Kohnen and D. Zagier, Heegner points and Derivatives of L-series. II, Math. Ann. 278 (1987), 497–562. CrossrefGoogle Scholar

  • [5]

    S. Choi and C. H. Kim, Rational period functions and cycle integrals in higher level cases, J. Math. Anal. Appl. 427 (2015), 741–758. CrossrefWeb of ScienceGoogle Scholar

  • [6]

    W. Duke, Ö. Imamoḡlu, and Á. Tóth, Cycle integrals of the j-function and mock modular forms, Ann. Math. 173 (2) (2011), 947–981. Web of ScienceCrossrefGoogle Scholar

  • [7]

    W. Duke, Ö. Imamoḡlu, and Á. Tóth, Rational period functions and cycle integrals, Abh. Math. Semin. Univ. Hambg. 80 (2010), no.2, 255-264. CrossrefGoogle Scholar

  • [8]

    M. Manickam and B. Ramakrishnan, On Shimura, Shintani and Eichler-Zagier Correspondences, Tran. Amer. Math. Soc. 352(2000), 2601–2617. CrossrefGoogle Scholar

About the article

Received: 2016-08-02

Accepted: 2017-01-03

Published Online: 2017-03-30


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 304–316, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0020.

Export Citation

© 2017 Choi and Kim. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in