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

  • SoYoung Choi and Chang Heon Kim EMAIL logo
Published/Copyright: March 30, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

For an odd and squarefree level N, Kohnen proved that there is a canonically defined subspace S κ + 1 2 n e w ( N ) ⊂ S κ + 1 2 ( N ) , and S κ + 1 2 n e w ( N ) and S 2 k n e w ( N ) are isomorphic as modules over the Hecke algebra. Later he gave a formula for the product a g ( m ) a g ( 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 S 2 k + ( p ) and S k + 1 2 ( 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

Γ 0 ( N ) := { a b c d ∈ S L 2 ( Z ) | c ≡ 0 ( m o d N ) }

be the Hecke subgroup of Γ(1):=SL2(ℤ), and Γ 0 + ( N ) be the group generated by the Hecke group Γ0(N) and the Fricke involution W N := 0 − 1 / N N 0 . If k is a positive integer, then we write S2k(N)(resp. S 2 k + ( N ) ) for the space of cusp forms of weight 2k on Γ0(N)(resp. Γ 0 + ( N ) ). We also denote by s k + 1 2 ( N ) the space of cusp forms of weight k + 1 2 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 S k + 1 2 ( N ) be the subspace of S k + 1 2 ( 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 s k + 1 2 ± , p ( N ) as the subspace of S k + 1 2 ( N ) consisting of forms whose n-th Fourier coefficients vanish for ( − 1 ) k n p = ∓ 1. In [1, Proposition 4] Kohnen defines, for each prime divisor p of N, a Hermitian involution w p , k + 1 2 N on S k + 1 2 ( N ) and shows that its (±)-eigenspace is equal to S k + 1 2 ± , p ( N ) . Thus we have an orthogonal decomposition

s κ + 1 2 ( N ) = s κ + 1 2 + , p ( N ) ⨁ s κ + 1 2 − , p ( N ) .

We then observe that

S k + 1 2 + , p ( p ) = S k + 1 2 ( p ) .

Let S 2 k n e w ( 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 S k + 1 2 n e w ( N ) ⊂ S k + 1 2 ( N ) , and S k + 1 2 n e w ( N ) and S 2 k n e w ( N ) are isomorphic as modules over the Hecke algebra. Later in [2, Theorem 3] he gave a formula for the product a g ( m ) a g ( 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 S 2 k + ( p ) and S k + 1 2 ( 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 κ ∈ 1 2 Z on some subgroup Γ of finite index in Γ(1) we denote their Petersson product by

〈 f , g 〉 = 〈 f , g 〉 Γ = 1 [ Γ ( 1 ) : Γ ] ∫ Γ ∖ H f ( z ) g ( z ) ¯ y κ − 2 d x d y ( x = R e z , y = I m z ) .

Let m be a positive integer such that (— 1)km = □ (mod 4p). Following [3], for z ∈ ℌ, we let P m + ( z ) ∈ S k + 1 2 ( p ) be a Poincaré series such that

〈 g , P m + ( z ) 〉 = a g ( m ) c ( m ) for all g = ∑ n ≥ 1 a g ( n ) q n ∈ S k + 1 2 ( p ) ,

where c ( m ) = Γ ( k − 1 2 ) i 4 p − 1 ( 4 π m ) k − 1 2 . 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 P k , p , α ( z ) ∈ S k + 1 2 ( p ) be the Poincaré series characterized by

〈 g , P k , p , α ( z ) 〉 = a g ( α ) c ( α ) for all g = ∑ n ≥ 1 a g ( n ) q n ∈ S k + 1 2 ( p ) .

Write

P k , p , α = P k , p , α + ⊕ P k , p , α − ∈ S k + 1 2 ( p ) ⨁ S k + 1 2 − , 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 P k , p , α + = P α + .

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

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

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

Let 𝔇 be the set of all discriminants, i.e.

D = { d ∈ Z | d ≡ 0 , 1 ( m o d 4 ) } .

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 = α β γ δ ∈ Γ 0 + ( 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 Γ 0 + ( 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

f k , N ( z ; D , D ′ ) := ∑ Q ∈ Q D D ′ , N χ D ( Q ) Q ( z , 1 ) k ( z ∈ H ) ,

where 𝓧d(Q)is the generalized genus character defined as follows: for Q = [a, b, c],

χ D ( Q ) := 0 i f ( a , b , c , D ) > 1 D r i f ( a , b , c , D ) = 1 , w h e r e [ a , b , c ] r e p r e s e n t s r , ( 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 Γ 0 + ( 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 κ ∈ 1 2 Z o n Γ 0 + ( p ) , we have

∫ Γ 0 ( p ) ∖ H f g ¯ y κ − 2 d x d y = 2 ∫ Γ 0 + ( p ) ∖ H f g ¯ y κ − 2 d x d y .

If d ∈ 𝔇 is positive and is not a perfect square, then the group

Γ 0 + ( p ) Q / { ± 1 } = { g ∈ Γ 0 + ( p ) | Q ∘ g = 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 f ∈ S 2 k + ( p ) and ∈ 𝓠d,p we define

r Q + ( f ) := ∫ c Q + f ( z ) d Q , k z

where c Q + = c Q + ( z 0 ) is the directed arc on SQ from z0 ∈ SQ to gQz0, and dQ,kz = Q(z, 1)k–1 dz. Then the value r Q + ( f ) is both independent of z0 ∈ SQ and a class invariant.

For each f ∈ S 2 k + ( p ) and integers D, m satisfying the following condition

(1) ( − 1 ) k m ∈ D , ( − 1 ) k D > 0 , m > 0 , | D | m ≠ ◻ , and | D | m ≡ ◻ ( m o d 4 p ) ,

we set

r k , p + ( f ; D , ( − 1 ) k m ) := ∑ Q ∈ Q | D | m , p / Γ 0 + ( p ) χ D ( Q ) r Q + ( f ) .

We also define for cusp forms f and g of weight 2k ∈ 2ℤ on Γ 0 + ( p ) ,

〈 f , g 〉 Γ 0 + ( p ) = 2 i p ∫ Γ 0 + ( p ) ∖ H f g ¯ y 2 k − 2 d x d y so that 〈 f , g 〉 Γ 0 + ( p ) = 〈 f , g 〉 Γ 0 ( p ) .

Proposition 1.4

For any f ∈ S 2 k + ( p ) and integers D, m satisfying (1) we have

〈 f , f k , p ( ; D , ( − 1 ) k m ) 〉 = 〈 f , f k , p ( ; D , ( − 1 ) k m ) 〉 Γ 0 + ( p ) = 2 i p π 2 k − 2 k − 1 2 − 2 k + 2 ( | D | m ) 1 2 − k r k , p + ( f ; D , ( − 1 ) k m ) .

Proposition 1.5

For D ∈ 𝔇 with (–1)k D > 0, we let

φ k , p ( z , τ ) := ∑ m ≥ 1 ( − 1 ) k m ≡ 0 , 1 ( 4 ) m k − 1 2 f k , p ( z ; D , ( − 1 ) k m ) e 2 π i m τ .

Then for fixed z, φk,p(z, τ) is an element of S k + 1 2 ( p ) .

Let w p := w p , k + 1 2 p be the Hermitian involution (with respect to τ) on the space S k + 1 2 ( p ) . We then have a decomposition

φ k , p ( z , τ ) = φ k , p + ( z , τ ) ⊕ φ k , p − ( z , τ ) ∈ S k + 1 2 ( p ) ⨁ S k + 1 2 − , p ( p ) .

Theorem 1.6

Let D ∈ 𝔇 with (–1)k D > 0 and D ≡ □ (mod 4p). Define for each f ∈ S 2 k + ( p ) , , the D-th Shintani lift

f | S + ∗ := i p c k , D ∗ 〈 f , φ k , p + ( , − τ ¯ ) 〉 ∈ S k + 1 2 ( p ) ,

with c k , D ∗ = i p ( − 1 ) [ k 2 ] 3 ( 2 π ) k ( k − 1 ) ! c ( | D | ) . If we write the Fourier expansion of g := f | S + ∗ as

g ( τ ) = ∑ m ≥ 1 ( − 1 ) k m ≡ ◻ ( m o d 4 p ) a g ( m ) e 2 π i m τ ,

then for a positive integer m satisfying p ∤ m and (–1)km ≡ □ .(mod 4p) one has

a g ( m ) = 2 r k , p + ( f ; D , ( − 1 ) k m ) .

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

φ k , p + ( z , τ ) = φ k , p ( z , τ ) + φ k , p ( z , τ ) | w p 2 .

Alternatively, we can find φ k , p + ( z , τ ) by means of Poincaré series as follows. Following [2, (3)], for z,τ ∈ ℌ and D ∈ 𝔇 with (–1)k D > 0, we let

Ω k , p ( z , τ ; D ) = i p c k , D − 1 ∑ m ≥ 1 ( − 1 ) k m ≡ 0 , 1 ( 4 ) m k − 1 2 ∑ t | p μ ( t ) D t t k − 1 f k , p t ( t z ; D , ( − 1 ) k m ) e 2 π i m τ

where c k , D = ( − 1 ) [ k 2 ] | D | − k + 1 2 π 2 k − 2 k − 1 2 − 3 k + 2 and μ(t) is the Möbius μ-function. By [2, (3)] or (5) one has

φ k , p ( z , τ ) = i p − 1 c k , D Ω k , p ( z , τ ; D ) + p k − 1 i p D p Ω k , 1 ( p z , τ ; D ) = ( − 1 ) [ k 2 ] 3 ⋅ ( 2 π ) k ( k − 1 ) ! ∑ n ≥ 1 n k − 1 ∑ d | n p ∤ d D d n d k P k , p , n 2 | D | / d 2 ( τ ) e 2 π i n z + p k − 1 ( D p ) ( − 1 ) [ k 2 ] 3 ⋅ ( 2 π ) k ( k − 1 ) ! ∑ n ≥ 1 n k − 1 ∑ d | n D d n d k P k , 1 , n 2 | D | / d 2 ( τ ) e 2 π i n p z .

This gives rise to the following theorem.

Theorem 1.8

Let D ∈ 𝔇 with (–1)kD > 0 and D ≡ □ (mod 4p). We have

(2) φ k , p + ( z , τ ) = ( − 1 ) [ k 2 ] 3 ⋅ ( 2 π ) k ( k − 1 ) ! ∑ n ≥ 1 n k − 1 ∑ d | n p ∤ d D d n d k P k , p , n 2 | D | / d 2 + ( τ ) e 2 π i n z + p k − 1 D p ( − 1 ) [ k 2 ] 3 ⋅ ( 2 π ) k ( k − 1 ) ! ∑ n ≥ 1 n k − 1 ∑ d | n D d n d k P k , 1 , n 2 | D | / d 2 + ( τ ) e 2 π i n p z .

Let D ∈𝔇with (–1)k D > 0 and D ≡ □ (mod 4p). Now we define for each g ∈ S k + 1 2 ( p ) , the D-th Shimura lift

g | S + := i p c k , D ∗ 〈 g , φ k , p + ( − z ¯ , τ ) 〉 τ ∈ S 2 k + ( p ) .

It then follows from (2) that

g | S + = i p c k , D ∗ ( − 1 ) [ k 2 ] 3 ⋅ ( 2 π ) k ( k − 1 ) ! ∑ n ≥ 1 n k − 1 ∑ d | n p ∤ d D d ( n d ) k a g ( n 2 | D | / d 2 ) c ( n 2 | D | / d 2 ) e 2 π i n z + p k − 1 i p c k , D ∗ D p ( − 1 ) [ k 2 ] 3 ⋅ ( 2 π ) k ( k − 1 ) ! ∑ n ≥ 1 n k − 1 ∑ d | n D d n d k 〈 g , P k , 1 , n 2 | D | / d 2 + ( τ ) 〉 τ e 2 π i n p z .

Theorem 1.9

(i) Our Shimura lift is Hecke equivariant, i.e. for each g ∈ S k + 1 2 ( p ) and a prime r(≠ p), we have

( g | S + ) | T ( r ) = ( g | T ( r 2 ) ) | S + ⋅

(ii) Our Shintani lift is Hecke equivariant, i.e. for each f ∈ S 2 k + ( p ) and a prime r(≠ p), we have

( f | S + ∗ ) | T ( r 2 ) = ( f | T ( r ) ) | S + ∗ .

Theorem 1.10

Let p be an odd prime. There exists a Hecke equivariant isomorphism

L ´ ( = L ´ p ) : S 2 k + ( p ) → S k + 1 2 ( 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 ) : S 2 k + ( p ) → S k + 1 2 ( p ) . be the Hecke equivariant isomorphism discussed in Theorem 1.10 and let {f1, . . . , ft} be a basis for S 2 k + ( 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

g j | S + = c j ( | D | ) f j

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

c j ( n ) ¯ = c ⋅ c j ( n )

for some nonzero constant c.

(iii) Let S + ∗ denote the D -th Shintani lift. Then

f j | S + ∗ = c j ( | D | ) ¯ 〈 f j , f j 〉 〈 g j , g j 〉 − 1 g j .

(iv) For each positive integer m such that (–1)km ≡ □ (mod 4p) and p ∤ m, we have

r k , p + ( f j , D , ( − 1 ) k m ) = 1 2 c j ( | D | ) ¯ c j ( m ) 〈 f j , f j 〉 〈 g j , g j 〉 − 1

(v) For each f ∈ S 2 k + ( p ) a n d g ∈ S k + 1 2 ( p ) , one has

〈 g | S + , f 〉 = 〈 g , f | S + ∗ 〉 ,

i.e. 𝓢+ and S + ∗ are adjoint.

Remark 1.12

(i) Let S k , p , D ∗ denote the Kohnen’s Shintani lift [2, (8)] and f be a newform in S 2 k + ( p ) . Then our Shintani lift f | S + ∗ is the projection of Kohnen’s Shintani lift f | S k , p , D ∗ o n S k + 1 2 ( 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 S k + 1 2 ( 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 S 2 k + ( p ) a n d S k + 1 2 ( p ) while we give it in Theorem 1.10.

(iv) The results in Theorem 1.11 hold for all Hecke eigenforms in S 2 k + ( p ) o r S k + 1 2 ( p ) But when we consider the spaces S 2 k + ( p ) a n d S k + 1 2 ( 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 f ∈ S k + 1 2 ( p ) a n d g ⊕ h ∈ S k + 1 2 ( p ) = S k + 1 2 ( p ) ⊕ s k + 1 2 − , p ( p ) ,

〈 f , g + h 〉 = 〈 f , g 〉 .

Write P k , p , α + ( τ ) = ∑ ( − 1 ) k l ≡ ◻ ( 4 p ) a P k , p , α + ( l ) e 2 π i l τ . We then have for each positive integer l with (–1)kl ≡ □ (mod 4p),

a P l + ( α ) c ( α ) = 〈 P l + , P k , p , α 〉 = 〈 P l + , P k , p , α + 〉 = 〈 P k , p , α , + P l + 〉 ¯ = c ( l ) a P k , p , α + ( l ) ¯ ,

so that

(3) a P k , p , α + ( l ) = c ( α ) c ( l ) a P l + ( α ) ¯ ,

which is zero unless (–1)k α ≡ □ (mod 4p). This proves the second assertion. To prove the first assertion we observe that

(4) a P l + ( α ) c ( α ) = 〈 P l + , P α + 〉 = 〈 P α + , P l + 〉 ¯ = a P α + ( l ) ¯ c ( l ) .

Combining (3) and (4) we get that a P k , p , α + ( l ) = a P α + ( 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

〈 P k , p , α − , P k , p , β − 〉 = 〈 P k , p , α − P α + , P k , p , β − P β + 〉 = c ( β ) a P k , p , α ( β ) − c ( α ) a P β + ( α ) ¯ = c ( β ) a P k , p , α ( β ) − c ( β ) a P α + ( β ) by (4) = c ( β ) a P k , p , α − ( β ) .

Thus

〈 P k , p , α , P k , p , β − 〉 = 〈 P α + + P k , p , α − , P k , p , β − 〉 = 〈 P k , p , α − , P k , p , β − 〉

= c ( β ) a P k , p , α − ( β ) = c ( α ) a P k , p , β − ( α ) ¯ = 0 since p ∤ α and ( − 1 ) k α p = 1.

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

Proof of Proposition 1.3. Let F p + be a fundamental domain of Γ 0 + ( p ) . Then F p + ⋃ ⋅ W p F p + is a fundamental domain of Γ0(p). Thus we see that

∫ Γ 0 ( p ) ∖ H f g ¯ y κ − 2 d x d y = ∫ F p + f g ¯ y κ − 2 d x d y + ∫ W p F p + f g ¯ y κ − 2 d x d y = ∫ F p + f g ¯ y κ − 2 d x d y + ∫ F p + f | κ W p g | κ W p ¯ y κ − 2 d x d y = 2 ∫ F p + f g ¯ y κ − 2 d x d y = 2 ∫ Γ 0 + ( p ) ∖ H f g ¯ y κ − 2 d x d y ,

as desired.

Proof of Proposition 1.4. For a Γ 0 + ( p ) -class 𝔘 of forms Q ∈ 𝓠|D|m,p, we define

f k , p ( z ; U ) = ∑ Q ∈ U Q ( z , 1 ) − k .

We then have f k , p ( z ; U ) ∈ S 2 k + ( p ) and

i p 〈 f , f k , p ( ; U ) 〉 Γ 0 + ( p ) = 2 ∫ Γ 0 + ( p ) ∖ H ∑ γ ∈ Γ Q ∖ Γ 0 + ( p ) ( Q ∘ γ ) ( z ¯ , 1 ) − k f ( z ) y 2 k − 2 d x d y = 2 ∫ Γ Q ∖ H Q ( z ¯ , 1 ) − k f ( z ) y 2 k − 2 d x d y ,

where Q is any element of 𝔘 and ΓQ is the stabilizer of Q in Γ 0 + ( p ) . By similar argument in [2, Proposition 7], we obtain

〈 f , f k , p ( ; U ) 〉 Γ 0 + ( p ) = 2 i p ( | D | m ) 1 2 − k ∫ 0 π sin 2 k − 2 ⁡ θ d θ ∫ c Q + f ( z ) d Q , k Z

and hence

〈 f , f k , p ( ; D , ( − 1 ) k m ) 〉 Γ 0 + ( p ) = 2 i p π 2 k − 2 k − 1 2 − 2 k + 2 ( | D | m ) 1 2 − k r k , p + ( f ; D , ( − 1 ) k m ) .

Proof of Proposition 1.5. We observe that

Ω k , p ( z , τ ; D ) = i p c k , D − 1 ∑ m ≥ 1 ( − 1 ) k m ≡ 0 , 1 ( 4 ) m k − 1 2 f k , p ( z ; D , ( − 1 ) k m ) e 2 π i m τ − i p c k , D − 1 ∑ m ≥ 1 ( − 1 ) k m ≡ 0 , 1 ( 4 ) m k − 1 2 D p p k − 1 f k , 1 ( p z ; D , ( − 1 ) k m ) e 2 π i m τ ∈ S k + 1 2 ( p ) .

Since

Ω k , 1 ( z , τ ; D ) = c k , D − 1 ∑ m ≥ 1 ( − 1 ) k m ≡ 0 , 1 ( 4 ) m k − 1 2 f k , 1 ( z ; D , ( − 1 ) k m ) e 2 π i m τ

belongs to S k + 1 2 ( 1 ) ⊆ S k + 1 2 ( p ) for fixed τ, one has

Ω k , 1 ( p z , τ ; D ) = c k , D − 1 ∑ m ≥ 1 ( − 1 ) k m ≡ 0 , 1 ( 4 ) m k − 1 2 f k , 1 ( p z ; D , ( − 1 ) k m ) e 2 π i m τ ∈ S k + 1 2 ( p ) .

Thus we obtain that

(5) φ k , p ( z , τ ) = ∑ m ≥ 1 ( − 1 ) k m ≡ 0 , 1 ( 4 ) m k − 1 2 f k , p ( z ; D , ( − 1 ) k m ) e 2 π i m τ = i p − 1 c k , D Ω k , p ( z , τ ; D ) + p k − 1 i p D p Ω k , 1 ( p z , τ ; D ) ∈ S k + 1 2 ( p ) ,

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,

a φ k , p + ( z , τ ) ( m ) = m k − 1 2 f k , p ( z ; D , ( − 1 ) k m ) .

Thus we have

g ( τ ) f | S + ∗ = i p c k , D ∗ 〈 f , φ k , p + ( , − τ ¯ ) 〉 = i p c k , D ∗ 〈 f , ∑ m ≥ 1 , p ∤ m ( − 1 ) k m ≡ ◻ ( 4 p ) m k − 1 2 f k , p ( ; D , ( − 1 ) k m ) e − 2 π i m τ ¯ 〉 + ∑ m ≥ 1 , p | m ( − 1 ) k m ≡ ◻ ( 4 p ) a g ( m ) e 2 π i m τ = i p c k , D ∗ ∑ m ≥ 1 , p ∤ m ( − 1 ) k m ≡ ◻ ( 4 p ) m k − 1 2 〈 f , f k , p ( ; D , ( − 1 ) k m ) 〉 e 2 π i m τ + ∑ m ≥ 1 , p | m ( − 1 ) k m ≡ ◻ ( 4 p ) a g ( m ) e 2 π i m τ = 2 ∑ m ≥ 1 , p ∤ m ( − 1 ) k m ≡ ◻ ( 4 p ) r k , p + ( f ; D , ( − 1 ) k m ) e 2 π i m τ + ∑ m ≥ 1 , p | m ( − 1 ) k m ≡ ◻ ( 4 p ) a g ( m ) e 2 π i m τ

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

(6) ( φ k , p | T ( r 2 ) ) + = φ k , p + | T ( r 2 )

and

(7) ( φ k , p | T ( r ) ) + = φ k , p + | T ( r ) .

Proof

We observe that

( φ k , p = φ k , p + ⊕ φ k , p − ) | T ( r 2 ) = φ k , p + | T ( r 2 ) ⊕ φ k , p − | T ( r 2 ) ∈ S k + 1 2 ( p ) ⨁ S k + 1 2 − , 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 g ∈ S k + 1 2 ( l ) ,

〈 g , Ω k , l ( − z ¯ , , D ) | T ( r 2 ) 〉 = { g | T ( r 2 ) , Ω k , l ( − z ¯ , , D ) } = ( 〈 g , Ω k , l ( − z ¯ , , D ) 〉 ) | T ( r ) b y [ 2 , p .241 ] = 〈 g , Ω k , l ( − z ¯ , , D ) | T ( r ) 〉 .

Thus one obtains

(8) Ω k , l ( − z ¯ , τ , D ) | T ( r 2 ) = Ω k , l ( − z ¯ , τ , D ) | T ( r )

where the operator T(r2)(resp. T(r))acts on a function with variable τ(resp. z). In particular,

(9) Ω k , 1 ( − p z ¯ , τ , D ) | T ( r 2 ) = Ω k , 1 ( − z ¯ , τ , D ) | T ( r 2 ) | V ( p ) = Ω k , 1 ( − z ¯ , τ , D ) | T ( r ) | V ( p ) = Ω k , 1 ( − z ¯ , τ , D ) | V ( p ) | T ( r ) = Ω k , 1 ( − p z ¯ , τ , D ) | T ( r )

where the operator V(p) acts on a function with variable z. Consequently, for g ∈ S k + 1 2 ( p ) ,

( g | S + ) | T ( r ) = ( 〈 g , φ k , p + ( − z ¯ , τ ) 〉 τ ) | T ( r ) = ( 〈 g , φ k , p ( − z ¯ , τ ) 〉 τ ) | T ( r ) = 〈 g , i p − 1 c k , D Ω k , p ( − z ¯ , τ , D ) 〉 | T ( r ) + 〈 g , p k − 1 c k , D D p Ω k , 1 ( − p z ¯ , τ , D ) 〉 | T ( r ) b y ( 5 ) = 〈 g , i p − 1 c k , D Ω k , p ( − z ¯ , τ , D ) | T ( r ) 〉 + 〈 g , p k − 1 c k , D D p Ω k , 1 ( − p z ¯ , τ , D ) | T ( r ) 〉 = 〈 g , i p − 1 c k , D Ω k , p ( − z ¯ , τ , D ) | T ( r 2 ) 〉 + 〈 g , p k − 1 c k , D D p Ω k , 1 ( − p z ¯ , τ , D ) | T ( r 2 ) 〉 b y ( 8 ) a n d ( 9 ) = 〈 g | T ( r 2 ) , φ k , p ( − z ¯ , τ ) 〉 = 〈 g | T ( r 2 ) , φ k , p + ( − z ¯ , τ ) 〉 = ( g | T ( r 2 ) ) | S + ⋅

Next we will prove the second assertion of Theorem 1.9. For f ∈ S2k(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

(10) 〈 f , Ω k , l ( , − τ ¯ , D ) 〉 = f | S ∗ f o r a l l f ∈ S 2 k ( l )

and

(11) ( f | T ( r ) ) | S ∗ = ( f | S ∗ ) | T ( r 2 ) f o r a l l p r i m e r ( ≠ l ) .

Thus for all f ∈ S2k(l), we have that

〈 f , Ω k , l ( , − τ ¯ , D ) | T ( r ) 〉 = 〈 f | T ( r ) , Ω k , l ( , − τ ¯ , D ) 〉 = 〈 f , Ω k , l ( , − τ ¯ , D ) 〉 | T ( r 2 ) b y ( 10 ) a n d ( 11 ) = 〈 f , Ω k , l ( , − τ ¯ , D ) | T ( r 2 ) 〉 ,

which implies that

Ω k , l ( z , − τ ¯ , D ) | T ( r ) = Ω k , l ( z , − τ ¯ , D ) | T ( r 2 ) .

In particular,

Ω k , 1 ( p z , − τ ¯ , D ) | T ( r ) = Ω k , 1 ( z , − τ ¯ , D ) | V ( p ) | T ( r ) = Ω k , 1 ( z , − τ ¯ , D ) | T ( r ) | V ( p ) = Ω k , 1 ( p z , − τ ¯ , D ) | T ( r 2 ) .

Hence we obtain that for all f ∈ S 2 k + ( p ) ,

c k , D ∗ i p ( f | S + ∗ ) | T ( r 2 ) = 〈 f , φ k , p + ( , − τ ¯ ) 〉 | T ( r 2 ) = 〈 f , φ k , p + ( , − τ ¯ ) | T ( r 2 ) 〉 = 〈 f , ( φ k , p ( , − τ ¯ ) | T ( r 2 ) ) + 〉 b y ( 6 ) = 〈 f , ( φ k , p ( , − τ ¯ ) | T ( r ) ) + 〉 = 〈 f , φ k , p + ( , − τ ¯ ) | T ( r ) 〉 b y ( 7 ) = 〈 f | T ( r ) , φ k , p + ( , − τ ¯ ) 〉 = c k , D ∗ i p ( f | T ( r ) ) | S + ∗ .

4 Proof of Theorem 1.10

Following [1, p.64 lines 7-9], let S k + 1 2 o l d ( p ) be the set of oldforms in S k + 1 2 ( p ) . Let S k + 1 2 o l d ( p ) := S k + 1 2 o l d ( p ) ∩ S k + 1 2 ( p ) .

Lemma 4.1

Define a map

Φ : S k + 1 2 ( 1 ) → S k + 1 2 o l d ( 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 f ∈ S k + 1 2 ( 1 ) . It then follows from [1, p.66 line 5] that f | wp is an oldform, too. Therefore we have an inclusion

{ f + f | w p | f ∈ S k + 1 2 ( 1 ) } ⊆ S k + 1 2 o 1 d ( p ) ∩ S k + 1 2 ( p ) = S k + 1 2 ol d ( p ) .

While [8, Theorem 5.11 in p.2612] says

S k + 1 2 ( p ) = S k + 1 2 n e w ( p ) ⨁ { f + f | w p | f ∈ S k + 1 2 ( 1 ) } ,

in [1] it is shown that

S k + 1 2 ( p ) = S k + 1 2 n e w ( p ) ⨁ S k + 1 2 o l d ( p ) .

Thus we have

(12) S k + 1 2 o l d ( p ) = { f + f | w p | f ∈ S k + 1 2 ( 1 ) } .

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 = ∑ n ≥ 1 a ( n ) q n ∈ S k + 1 2 ( 1 ) , then we have

0 = f + f | w p = ∑ n ≥ 1 a ( n ) q n + ∑ n ≥ 1 ( − 1 ) k n p a ( n ) + p k a ( n / p 2 ) q n b y [ 1 , ( 44 ) ] .

Comparing the coefficients of q p 2 m 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

S k + 1 2 ( p ) = S k + 1 2 ( 1 ) ⨁ S k + 1 2 ( 1 ) | U ( p 2 ) ⨁ S k + 1 2 n e w ( 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 | w p = p 1 − k ( − f | U ( p 2 ) + f | T k + 1 2 , 1 ( p 2 ) .

Thus for any prime r(≠ p), one has

Φ ( f ) | T ( r 2 ) = ( f + f | w p ) | T ( r 2 ) = f | T ( r 2 ) + ( f | w p ) | T ( r 2 ) = f | T ( r 2 ) + ( f | T ( r 2 ) ) | w p s i n c e U ( p 2 ) a n d T ( p 2 ) c o m m u t e w i t h T ( r 2 ) = Φ ( f | T ( r 2 ) .

Let Φ 1 : S 2 k ( 1 ) → S k + 1 2 ( 1 ) be the Kohnen-Shimura isomorphism which is Hecke equivariant, and we let S 2 k ( 1 ) → S 2 k + o l d ( 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 := Φ ∘ Φ 1 ∘ Φ 2 − 1 defines a Hecke equivariant isomorphism from S 2 k + o l d ( p ) to S k + 1 2 + o l d ( p ) . Meanwhile, according to [1, page 71] there exists a Hecke equivariant isomorphism from S 2 k + n e w ( p ) to S k + 1 2 n e w ( p ) . Now we see that the map

L ´ : S 2 k + ( p ) ( = S 2 k + o l d ( p ) ⨁ S 2 k + n e w ( p ) ) → S k + 1 2 ( p )

defined by

L ´ ( f ⊕ g ) = 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),

( g j | S + ) | T ( r ) = ( g j | T ( r 2 ) ) | S + = L ´ ( f j | T ( r ) ) | S + = λ j ( r ) L ´ ( f j ) | S + = λ j ( r ) g j | S + ,

for some λj (r)∈ ℂ By multiplicity one theorem we have

(13) g j | S + = c f j for some c ∈ C .

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 ( g j | T ( r 2 ) ) = L ´ − 1 ( g j ) | T ( r ) = f j | T ( r ) = λ j ( r ) f j = L ´ − 1 ( λ j ( r ) g j ) ,

so that

(14) g j | T ( r 2 ) = λ j ( r ) g j

Thus we have

g j ( − τ ¯ ) ¯ | T ( r 2 ) = ( g j | T ( r 2 ) ) ( − τ ¯ ) ¯ = λ j ( r ) g j ( − τ ¯ ) ¯

and therefore

L ´ − 1 ( g j ( − τ ¯ ) ¯ ) | T ( r ) = L ´ − 1 ( g j ( − τ ¯ ) ¯ | T ( r 2 ) ) = λ j ( r ) L ´ − 1 ( g j ( − τ ¯ ) ¯ ) .

This implies that

L ´ − 1 ( g j ( − τ ¯ ) ¯ ) = c f j = c L ´ − 1 ( g j ) for some nonzero c ∈ C .

Hence we have

g j ( − τ ¯ ) ¯ = c g j ( τ )

and therefore

c j ( n ) ¯ = c ⋅ c j ( n ) .

(iii) We note that g1, . . . , gt are mutually orthogonal with respect to Petersson scalar product. Write

φ k , p + ( z , τ ) = ∑ j = 1 t h j ( z ) g j ( τ ) .

We then have

φ k , p + ( − z ¯ , τ ) = ∑ j = 1 t h j ( − z ¯ ) g j ( τ ) ,

so that

g j | S + = 〈 g j , φ k , p + ( − z ¯ , τ ) 〉 τ = h j ( − z ¯ ) ¯ 〈 g j , g j 〉 .

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

h j ( z ) = c j ( | D | ) ¯ 〈 g j , g j 〉 ¯ f j ( − z ¯ ) ¯ = c j ( | D | ) ¯ 〈 g j , g j 〉 ¯ f j ( z ) ,

since fj has real Fourier coefficients. Hence we get that

φ k , p + ( z , τ ) = ∑ j = 1 t c j ( | D | ) ¯ 〈 g j , g j 〉 ¯ f j ( z ) g j ( τ ) ,

and therefore

c k , D ∗ i p f j | S + ∗ = 〈 f j ( z ) , φ k , p + ( z , − τ ¯ ) 〉 z = c j ( | D | ) 〈 f j , f j 〉 〈 g j , g j 〉 g j ( − τ ¯ ) ¯ = c j ( | D | ) 〈 f j , f j 〉 〈 g j , g j 〉 g j by the assertion (ii) .

(iv) Recalling from Theorem 1.6 that

F j := f j | S + ∗ = ∑ m ≥ 1 , p ∤ m ( − 1 ) k m ≡ ◻ ( 4 p ) 2 r k , p + ( f j ; D , ( − 1 ) k m ) e 2 π i m τ + ∑ m ≥ 1 , p | m ( − 1 ) k m ≡ ◻ ( 4 p ) a F j ( m ) e 2 π i m τ ,

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

〈 g i | S + , f j 〉 = 〈 g i , f j | S + ∗ 〉 .

Indeed we obtain that

〈 g i | S + , f j 〉 = 〈 c j ( | D | ) f i , f j 〉 = c j ( | D | ) δ i j 〈 f i , f j 〉

and by the assertion (iii)

〈 g i , f j | S + ∗ 〉 = 〈 g i , g j 〉 c j ( | D | ) 〈 f j , f j 〉 〈 g j , g j 〉 = c j ( | D | ) δ i j 〈 f i , f j 〉 .

More generally if we write g = ∑ i = 1 t a i g i and f = ∑ j = 1 t b j f j , then

〈 g | S + , f 〉 = 〈 ∑ i = 1 t a i g i | S + , ∑ j = 1 t b j f j 〉 = ∑ i , j a j b j ¯ 〈 g i | S + , f j 〉 = ∑ i , j a j b j ¯ 〈 g i , f j | 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).

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Received: 2016-8-2
Accepted: 2017-1-3
Published Online: 2017-3-30

© 2017 Choi and Kim

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/math-2017-0020
Keywords for this article
Modular forms; Shintani lifts; Shimura lifts
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