Abh. Math. Semin. Univ. Hambg. (2018) 88:67–86 https://doi.org/10.1007/s12188-017-0178-1

Period of the adelic Ikeda lift for U(m, m) Hidenori Katsurada1

Received: 20 October 2016 / Published online: 17 April 2017 © The Author(s) 2017

Abstract We give a period formula for the adelic Ikeda lift of an elliptic modular form f for U (m, m) in terms of special values of the adjoint L-functions of f. This is an adelic version of Ikeda’s conjecture on the period of the classical Ikeda lift for U (m, m). Keywords Period · Adelic Hermitian Ikeda lift Mathematics Subject Classification 11F55 · 11F67

1 Introduction In a previous paper [5], we proved Ikeda’s conjecture on the period of the classical Ikeda √ lift for U (m, m). In this paper we consider its adelic version. Let K = Q( −D) be an imaginary quadratic field with discriminant −D. Let h = h K be the class number of K , and χ the Kronecker character corresponding to the extension K /Q. Let U (m) = U (m, m) be the unitary group attached to the extension K /Q. For the precise definition of U (m) , see Sect. 2. Let m = 2n or 2n + 1, and let k be a non-negative integer. Put S(m,k) = S2k+1 (Γ0 (D), χ) or S(m,k) = S2k (S L 2 (Z)) according as m = 2n or 2n+1. Then for a primitive form f in S(m,k) satisfying a certain condition, let Li f t (m) ( f ) be the lift of f to the space of automorphic forms on the adele group U (m) (A) constructed by Ikeda [2]. We call Li f t (m) ( f ) the adelic Ikeda lift of f for U (m, m). Then, in this paper, we prove the following (cf. Theorem 2.2): (P.1) The period Li f t (m) ( f ), Li f t (m) ( f ) is expressed as L(1, f, Ad)

m 

L(i, f, Ad, χ i−1 )L(i, χ i )

i=2

Communicated by Jens Funke.

B 1

Hidenori Katsurada [email protected] Muroran Institute of Technology, 27-1 Mizumoto, Muroran 050-8585, Japan

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up to elementary factor, where L(s, f, Ad, χ i−1 ) is the twist of the adjoint L-function of f by χ i−1 , and L(i, χ i ) is the Dirichlet L-function for χ i . We note this result was proved in [7] in the case m = 2 and D is prime. Therefore, our result is new even in the case m = 2. To prove Theorem 2.2, we consider a refinement of the main result in [5]. For r = 1, . . . , h, let Im ( f )r be the r -th component of Li f t (m) ( f ) (cf. Section 2). In particular, we write Im ( f ) = Im ( f )1 and call it the classical Ikeda lift or simply the Ikeda lift of f for U (m, m). Then, we prove the following (cf. Theorem 2.1): (P.2) The period Im ( f )r , Im ( f )r  is expressed as L(1, f, Ad)

m 

L(i, f, Ad, χ i−1 )L(i, χ i )

i=2

up to elementary factor. We note that the result for r = 1 was given in [5]. Hence our present result can be regarded as a refinement of the previous result. We also note that the elementary factor in (P.2) is a little different from the elementary factor in (P.1) in general. The method we use is similar to that of a main result of [5] and to consider the Dirichlet series Rr (s, Im ( f )r ) of Rankin–Selberg type associated to Im ( f )r . First we can express its residue at 2k + 2n in terms of the period of Im ( f )r (cf. Corollary to Proposition 3.1). We then give an explicit formula for Rr (s, Im ( f )r ) (cf. Theorems 5.1 and 5.2), and compare its residue with Corollary to Proposition 3.1. To prove Theorems 5.1 and 5.2, in Sect. 4, we reduce our computation to a computation of certain formal power series Hm, p (d; X, Y, t) in t associated with local Siegel series similarly to [5] (cf. Theorem 4.1). To prove Theorem 4.1, we use a refined version of the mass formula for the special unitary group of a Hermitian matrix in [[3], Proposition 3.6]. An explicit formula for Hm, p (d; X, Y, t) was given in [[5], Theorems 5.5.2, 5.5.3, 5.5.4. Thus we obtain Theorem 2.1. Since the period Li f t (m) ( f ), Li f t (m) ( f ) can be expressed as h −1

h  Im ( f )r , Im ( f )r , r =1

we can prove Theorem 2.2 by Theorem 2.1. We also give another proof to Theorem 2.2. The idea is similar to above and is to give an  Li f t (m) ) of Li f t (m) in terms of certain explicit formula of the Rankin–Selberg series R(s, m, p (ι; X, Y, t) and H m, p (; X, Y, t) (cf. Theorem 4.2). The difference is that power series H we here use a modification of the mass formula for the unitary group of a Hermitian matrix m, p (ω; X, Y, t) by using in [[3], Theorem 3.6]. In Sect. 5 we give an explicit formula for H a result in [5] (cf. Theorems 6.2 and 6.3). In Sect. 7, by using Theorems 6.2 and 6.3, we  Li f t (m) ( f )) (cf. Theorems 7.1 and 7.2) and by immediately get an explicit formula of R(s, taking the residue of it at 2k + 2n we prove the Theorem 2.2. We note that our main result in this paper and that in [5] are very similar, but we cannot derive our present results from the previous one in the case h is greater than 1. We also note that this type of the period relation can be applied to a problem concerning congruence between the Ikeda lift and non-Ikeda lift (cf. [7]). The author was partially supported by JSPS KAKENHI Grant Numbers 25247001 and 16H03919. The author thanks T. Ikeda for many useful comments. The author also thanks the referee for useful comments. Notation. Let R be a commutative ring. We denote by R × and R ∗ the semigroup of non-zero elements of R and the unit group of R, respectively. We denote by Mmn (R) the set of (m, n)-matrices with entries in R. In particular put Mn (R) = Mnn (R). Put G L m (R) =

123

Period of the adelic Ikeda lift for U (m, m)

69

{A ∈ Mm (R) | det A ∈ R ∗ }, where det A denotes the determinant of a square matrix A. Let K 0 be a field, and K a quadratic extension of K 0 , or K = K 0 ⊕ K 0 .In the latter case, we regard K 0 as a subring of K via the diagonal embedding. If K is a quadratic extension of K 0 , let ρ be the non-trivial automorphism of K over K 0 , and if K = K 0 ⊕ K 0 , let ρ be the automorphism of K defined by ρ(a, b) = (b, a) for (a, b) ∈ K . We sometimes write x instead of ρ(x) for x ∈ K in both cases. Let R be a subring of K . For an (m, n)-matrix X = (xi j )m×n write X = (xi j )m×n and X ∗ = t X , and for an (m, m)-matrix A, we write A[X ] = X ∗ AX. Let Her n (R) denote the set of Hermitian matrices of degree n with entries in R, that is the subset of Mn (R) consisting of matrices X such that X ∗ = X. Then an Hermitian matrix A of degree n with entries in K is said to be semi-integral over R if tr(AB) ∈ K 0 ∩ R  for any B ∈ Sn (R), where tr denotes the trace of a matrix. We denote by H er n (R) the set of semi-integral matrices of degree n over R. For a subset S of Mn (R) we denote by S × the subset of S consisting of non-degenerate matrices. If S is a subset of Her n (C) with C the field of complex numbers, we denote by S + the subset of S consisting of positive definite matrices. G L n (R) acts on the set Her n (R) in the following way: G L n (R) × Her n (R) (g, A) −→ g ∗ Ag ∈ Her n (R). Let G be a subgroup of G L n (R). For a subset B of Her n (R) stable under the action of G we denote by B/G the set of equivalence classes of B with respect to G L n (R). We sometimes identify B/G with a complete set of representatives of B/G. We abbreviate B/G L n (R) as B/ ∼ if there is no fear of confusion. Two symmetric matrices A and A with entries in R are called equivalent over R with each other and write A ∼ R A if there is an element X of G L n (R ) such that A = A[X ]. We also write A if there is no fear of confusion. For  A ∼ X O square matrices X and Y we write X ⊥Y = . O Y √ We put e(x) = exp(2π −1x) for x ∈ C, and for a prime number p we denote by e p (∗) the continuous additive character of Q p such that e p (x) = e(x) for x ∈ Z[ p −1 ]. For a prime number p we denote by ord p (∗) the additive valuation of Q p normalized so that ord p ( p) = 1, and put |x| p = p −ord p (x) . Furthermore we denote by |x|∞ the absolute value of x ∈ C.

2 Period of the Ikeda lift for U(m, m) Throughout the paper, we fix an imaginary quadratic extension K of Q with the discriminant (m) = U (m, m) be the −D, and denote by O the ring of integersin K . For such  a K let U Om −1m , where 1m denotes the unit matrix unitary group defined in Sect. 1. Put Jm = 1m O m of degree m. Then U (m) (Q) = {M ∈ G L 2m (K ) | Jm [M] = Jm }.

Put Γ (m) = U (m) (Q) ∩ G L 2m (O). Let Hm be the Hermitian upper half-space defined by  1 Hm = Z ∈ Mn (C) | √ (Z − Z ∗ ) is positive definite . 2 −1

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Let l be an integer. For a subgroup Γ of U (m) (Q) which is commensurable with Γ (m) and a character χ of Γ, we denote by Ml (Γ, χ) the space of holomorphic modular forms of weight l with character χ for Γ. We denote by Sl (Γ, χ) the subspace of Ml (Γ, χ) consisting of cusp forms. In particular, if χ is the character of defined by χ(γ ) = (det γ )−l for γ ∈ , we write M2l (Γ, χ) as M2l (Γ, det−l ), and so on. For a positive integer N we define two (m) subgroups Γ0 (N ) and Γ (m) (N ) by



(m) B ∈ Γ (m) ∩ S L (K ) | C ≡ O mod N , Γ0 (N ) = CA D 2m m and Γ (m) (N ) = {g ∈ Γ (m) ∩ S L 2m (K ) | g ≡ 12m mod N }.  B ∈ Γ (m) ∩S L 2m (K ). It is well-known that det A, det B, det C and det D belong to Z for CA D  B (m) Hence for a Dirichlet character η mod N we can define the character of Γ0 (N ) by CA D → η(det D), which will be denoted also by η. For a subgroup Γ of U (m) (Q) commensurable with Γ (m) we put  (det Y )−2m d X dY, v(Γ ) = Γ \Hm

Z +t Z

Z −t Z

where X = 2 , and Y = 2√−1 . Moreover for two cusp forms F and G of weight l with respect to Γ with character χ we define the Petersson scalar product F, G by  F, G = v(Γ (m) )v(Γ )−1 F(Z )G(Z )(det Y )l−2m d X dY, Γ \Hm

We call F, F the period of F. We note that we have v(Γ (m) )v(Γ )−1 = [Γ (m) : O∗ Γ ]−1 if Γ is a congruence subgroup of Γ (m) . Let A be the adele ring of Q, and A f the non-archimedian factor of A. Let h = h K be a class number of K . Let G (m) = Res K /Q (G L m ), and G (m) (A) be the adelization of G (m) . Moreover put C (m) = p G L m (O p ). Let U (m) (A) be the adelization  (m) of U (m) . We define the compact subgroup K0 of U (m) (A f ) by U (m) (A) ∩ p G L 2m (O p ), where p runs over all rational primes. Then we have U (m) (A) =

h 

(m)

U (m) (Q)γr K0 U (m) (R)

r =1

with some subset {γ1 , . . . , γh } of

U (m) (A

f ).



γr =

We can take γr as  tr 0 , 0 tr∗−1

where {tr }rh=1 = {(tr, p )}rh=1 is a certain subset of G (m) (A f ) such that t1 = 1, and G (m) (A) =

h 

G (m) (Q)tr G (m) (R)C (m) .

r =1 (m)

Put Γr = U (m) (Q) ∩ γr K0 γr−1 U (m) (R). For an element (F1 , ..., Fh ) ∈ det −l ), we define (F1 , ..., Fh ) by (F1 , ..., Fh ) (g) = Fr (xi) j (x, i)−2l (det x)l

123

h

(m) r =1 M2l (Γr ,

Period of the adelic Ikeda lift for U (m, m)

71

for g = uγr xκ with u ∈ U (m) (Q), x ∈ U (m) (R), κ ∈ K0 . We denote by Ml (U (m) (Q)\U (m) (A), det −l ) the space of automorphic forms obtained in this way. We also put S2l (U (m) (Q)\U (m) (A), det −l ) = {(F1 , ..., Fh ) | Fr ∈ S2l (Γr(m) , det −l )}.

For elements F = (F1 , ..., Fh ) and G = (F1 , ..., Fh ) of S2l (U (m) (Q)\U (m) (A), det −l ) we define F, G by F, G = h −1

h  Fr , G r . r =1

We can define the Hecke operators which act on the space S2l (U (m) (Q)\U (m) (A), det −l ). For the precise definition, see [2].  m (O) be the set of semi-integral Hermitian matrices over O of degree m as in the Let Her  Notation. We note that √ A belongs to Her m (O) if and only if its diagonal components are rational integers and −D A ∈ Her m (O). For a non-degenerate Hermitian matrix B with entries in K of degree m, put γ (B) = (−D)[m/2] det B. For a prime number p put K p = K ⊗ Q p , and O p = O ⊗ Z p . For x ∈ K p we put ν K p (x) = ord p (x x), and |x| K p = |x x| p . Furthermore put |x| K ∞ (x) = |x x|∞ for x ∈ C. For a non-degenerate Hermitian matrix B with entries in K p of degree m, put γ p (B) =  m (O p ) be the set of semi-integral matrices over O p of degree m (−D)[m/2] det B. Let Her as in the Notation. We put ξ p = 1, −1, or 0 according as K p = Q p ⊕ Q p , K p is an unramified quadratic extension of Q p , or K p is a ramified quadratic extension of Q p . Now  m (O p )× we define the local Siegel series b p (T, s) by for T ∈ Her  b p (T, s) = e p (tr(T R)) p −ord p (μ p (R))s , R∈Her m (K p )/Her m (O p )

where μ p (R) = [R Omp + Omp : Omp ]1/2 . Let F p (T, X ) be the polynomial in X arising from the local Siegel series defined in [5]. We then define a Laurent polynomial F˜ p (T, X ) as F˜ p (T, X ) = X −ord p (γ p (T )) F p (T, p −m X 2 ). We remark that we have if m is odd, F˜ p (T, X −1 ) = F˜ p (T, X ) −1 if m is even, F˜ p (T, X ) = (−D, γ p (T )) p F˜ p (T, X ) −1 ˜ ˜ if m is even and p  D F p (T, ξ p X ) = F p (T, X ) (cf. [2]). Here (a, b) p is the Hilbert symbol of a, b ∈ Q× p . Hence we have X ord p (γ p (T )) F p (T, p −m X −2 ). F˜ p (T, X ) = (−D, γ p (T ))m−1 p Now we put   er m (O p ) for any p}. H er m (O)r+ = {T ∈ Her m (K )+ | tr,∗ p T ti, p ∈ H Let k be a non-negative integer. First let m = 2n be a positive even integer and let f (z) =

∞ 

a(N )e(N z)

N =1

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be a primitive form in S2k+1 (Γ0 (D), χ). For a prime number p not dividing D let α p ∈ C −k −k such that α p + χ( p)α −1 p = p a( p), and for p | D put α p = p a( p). Then for the i Kronecker character χ we define Hecke’s L-function L(s, f, χ ) twisted by χ i as

−1 

−s+k L(s, f, χ i ) = χ( p)i+1 ) (1 − α p p −s+k χ( p)i )(1 − α −1 p p pD



×



p|D (1 − α p p

−s+k )−1

−1 −s+k )−1 p|D (1 − α p p

if i is even if i is odd.

In particular, if i is even, we sometimes write L(s, f, χ i ) as L(s, f ) as usual. Moreover for r = 1, ..., h we define a Fourier series  a Im ( f )r (T )e(tr(T Z )), Im ( f )r (Z ) =  m (O )r+ T ∈Her

where a I2n ( f )r (T ) = |γ (T )|k



p (tr,∗ p T tr, p , α −1 | det(tr, p ) det(tr, p )|np F p ).

p

Next let m = 2n + 1 be a positive odd integer and let f (z) =

∞ 

a(N )e(N z)

N =1

be a primitive form in S2k (S L 2 (Z)). For a prime number p let α p ∈ C such that α p + α −1 p = p −k+1/2 a( p). Then we define Hecke’s L-function L(s, f, χ i ) twisted by χ i as

−1 

−s+k−1/2 (1 − α p p −s+k−1/2 χ( p)i )(1 − α −1 χ( p)i ) . L(s, f, χ i ) = p p p

In particular, if i is even we write L(s, f, χ i ) as L(s, f ) as usual. Moreover for r = 1, ..., h we define a Fourier series  a I2n+1 ( f )r (T )e(tr(T Z )), I2n+1 ( f )r (Z ) =  2n+1 (O )r+ T ∈Her

where a I2n+1 ( f )r (T ) = |γ (T )|k−1/2



n+1/2  ∗ F p (tr, p T tr, p , α −1 p ).

| det(tr, p ) det(tr, p )| p

p

Then Ikeda [2] showed the following: Let m = 2n or 2n + 1. Let f be a primitive form in S2k+1 (Γ0 (D), χ) or in S2k (S L 2 (Z)) (m) according as m = 2n or m = 2n + 1. Moreover let Γr be the subgroup of U (m) defined as (m) above. Then Im ( f )r (Z ) is an element of S2k+2n (Γr , det−k−n ) for any r . It follows from the above result that we can define an element (Im ( f )1 , ..., Im ( f )h ) of S2k+2n (U (m) (Q)\U (m) (A), det −k−n ), which we write Li f t (m) ( f ). Then he also showed the following: Let m = 2n or 2n + 1, and f be as above. We assume the following condition (*) f does not come from a Hecke character of some imaginary quadratic field if m = 2n with n odd.

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Then Li f t (m) ( f ) is a Hecke eigenform in S2k+2n (U (m) (Q)\U (m) (A), det−k−n ) whose standard L-function in the sense of Shimura [8] is m 

L(s + k + n − i + 1/2, f )L(s + k + n − i + 1/2, f, χ).

i=1

Remark In [2], Ikeda defined the standard L function in the automorphic representation theoretic view point, and gave an explicit form of the Euler factor of the standard L-function of Li f t (m) ( f ) only for good primes. However, if we define the standard L function in the sense of Shimura [8], it is not hard to get an explicit form of the Euler factor of it for any prime. We call Li f t (m) ( f ) the Ikeda lift of f for U (m) . To state our main result, put C (s) = 2(2π)−s (s). For an integer i let L(s, χ i ) = ζ (s) or L(s, χ) according as i is even or odd, where ζ (s) and L(s, χ) are Riemann’s zeta function, and Dirichlet L-function for χ, respectively, and put (s, χ i ) = C (s)L(s, χ i ).  For a primitive form f in S2k+1 (Γ0 (D), χ), we define the adjoint L-function L(s, f, Ad) and its twist L(s, f, Ad, χ) by χ as

−1  

−s −s (1 − α 2p χ( p) p −s )(1 − α −2 (1 − p −s )−1 , L(s, f, Ad) = p χ( p) p )(1 − p ) p|D

pD

and L(s, f, Ad, χ) =

−1 

−s −s (1 − α 2p p −s )(1 − α −2 p )(1 − χ( p) p ) . p p

For a primitive form f in S2k (S L 2 (Z)), we define the adjoint L-function L(s, f, Ad) and its twist L(s, f, Ad, χ) by χ as

−1 

−s −s L(s, f, Ad) = (1 − α 2p p −s )(1 − α −2 , p p )(1 − p ) p

and L(s, f, Ad, χ) =

−1 

−s −s (1 − α 2p χ( p) p −s )(1 − α −2 . p χ( p) p )(1 − χ( p) p ) p

Let f be a primitive form in S2k+1 (Γ0 (D), χ) or in S2k (S L 2 (Z)) according as m = 2n or m = 2n + 1. We then put  i is even (s, f, Ad, χ i ) = C (s) C (s + 2k − m) L(s, f, Ad)  L(s, f, Ad, χ) i is odd. Let Q D be the set of prime divisors of D. For each prime q ∈ Q D , put Dq = q ordq (D) . We define a Dirichlet character χq by  χ(a ) if (a, q) = 1 , χq (a) = 0 if q|a

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where a is an integer such that a ≡ a mod Dq and a ≡ 1 mod D Dq−1 .   = For a subset Q of Q D put χ Q = q∈Q χq and χ Q q∈Q D ,q ∈Q / χq . Here we make the = χ if Q is the empty set. Let convention that χ Q = 1 and χ Q f (z) =

∞ 

c f (N )e(N z)

N =1

be a primitive form in S2k+1 (Γ0 (D), χ). Then there exists a primitive form f Q (z) =

∞ 

c f Q (N )e(N z)

N =1

such that / Q c f Q ( p) = χ Q ( p)c f ( p) for p ∈ and c f Q ( p) = χ Q ( p)c f ( p) for p ∈ Q.

Let tr = (tr, p ) p ∈ G(m) (A f ) be as defined before. Let   C(tr ) := |N K p /Q p (det tr, p )| p , r (Q) = (−D, C(tr ))q , p

q∈Q

and



ηn,r ( f ) =

χ Q ((−1)n )r (Q).

Q⊂Q D fQ= f

Then our main result in this paper is: Theorem 2.1 (1) Let m = 2n be a positive even integer. For a primitive form f in S2k+1 (Γ0 (D), χ), we have I2n ( f )r , I2n ( f )r  = 2−4nk−4n ×

2n 

2 −4n+2

D 2nk+5n

(i, f, Ad, χ i−1 ) 

i=1

2 −3n/2−1/2

2n 

ηn,r ( f )

(i, χ i ). 

i=2

(2) Let m = 2n + 1 be a positive odd integer. For a primitive form f in S2k (S L 2 (Z)), we have I2n+1 ( f )r , I2n+1 ( f )r  = 2−2(2n+1)k−4n ×

2n+1 

2 −6n

D 2nk+5n

(i, f, Ad, χ i−1 ) 

i=1

For a primitive form f in S2k+1 (Γ0 (D), χ), let  1 + χ((−1)n ) if f Q D = f  ηn ( f ) = 1 if f Q D  = f.

123

2 +5n/2

2n+1  i=2

(i, χ i ). 

Period of the adelic Ikeda lift for U (m, m)

75

We note that  ηn ( f ) coincides with ηn ( f ) in [[2], Conjecture 17.6] in the case h = 1, but it is not in general. Then, by the genus theory of quadratic fields, we have h 

ηn,i ( f ) = h ηn ( f ).

i=1

Then we have Theorem 2.2 (1) Let m = 2n be a positive even integer, and f a primitive form in S2k+1 (Γ0 (D), χ). Then we have Li f t (2n) ( f ), Li f t (2n) ( f ) = 2−4nk−4n ×

2n 

2 −4n+2

D 2nk+5n

(i, f, Ad, χ i−1 )

i=1

2 −3n/2−1/2

2n 

 ηn ( f )

(i, χ i ). 

i=2

(2) Let m = 2n + 1 be a positive odd integer, f a primitive form f in S2k (S L 2 (Z)). Then we have Li f t (2n+1) ( f ), Li f t (2n+1) ( f ) = 2−2(2n+1)k−4n ×

2n+1 

2 −6n

D 2nk+5n

(i, f, Ad, χ i−1 ) 

i=1

2 +5n/2

2n+1 

(i, χ i ). 

i=2

Now put L(i, f, Ad, χ i−1 ) =

(i, f, Ad, χ i−1 )   f, f 

for i = 1, ..., m (2i, χ 2i ), L(2i, χ 2i ) =  and (2i + 1, χ 2i+1 )D 2i+1/2 L(2i + 1, χ 2i+1 ) =  for an integer i ≥ 1. We note that  2k+1  −1 2 q|D (1 + q ) if f ∈ S2k+1 (Γ0 (D), χ) L(1, f, Ad) = 2k if f ∈ S2k (S L 2 (Z)). 2 Hence we obtain the following: Theorem 2.3 Let the notation be as above. Then we have m  Li f t (m) ( f ), Li f t (m) ( f ) βn,k γn,k = 2 D L(i, f, Ad, χ i−1 )L(i, χ i )  f, f m i=2    ηn ( f ) q|D (1 + q −1 ) if m = 2n × 1 if m = 2n + 1,

where βn,k and γn,k are integers depending on n and k. It is well-known that L(i, χ i ) is a rational number for any positive integer i. Moreover L(i, f, Ad, χ i−1 ) is an algebraic number and belongs to the Hecke field Q( f ) for i = 2, ...., k where k = 2k or 2k − 1 according as if m is even or odd (cf. Shimura [8,9]). Hence we obtain

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H. Katsurada

Corollary In addition to the above notation and the assumption, assume that m ≤ 2k or Li f t (m) ( f ), Li f t (m) ( f ) m ≤ 2k − 1 according as m is even or odd. Then is algebraic,  f, f m and in particular it belongs to Q( f ). As in the symplectic case [1,4], we propose the following conjecture: m L(i, f, Ad, χ i−1 ) Conjecture A Let m and f be as above. Then a prime divisor of i=2 (m) gives a congruence between Li f t ( f ) and a Hecke eigenform in S2k+2n (U (m) (Q)\U (m) (A), det −k−n ) not coming from the Ikeda lift.

3 Rankin–Selberg convolution product To prove Theorems 2.1 and 2.2, we rewrite them in terms of the residues of the Rankin–Selberg convolution products of Im ( f )r and Li f t (m) ( f ). As in Sect. 1, we decompose G (m) (A) as follows: G (m) (A) =

h 

G (m) (Q)tr G (m) (R)C (m) ,

r =1

where {tr }rh=1 = {(tr, p )}rh=1 is a certain subset of G (m) (A f ) such that t1 = 1. Let (m)  er m (O)r+ and the others be as in Sect. 2. Let Γr H  a F (A)e(tr(Az) F(z) =  m (O )r+ A∈Her (m)

be an element of S2l (Γr Rr (s, F) for F by

Rr∗ (s, F) =

, det−l ). We then define the Rankin–Selberg series Rr∗ (s, F) and





| det(tr, p )|s−2l Kp

p

+

a F (A)a F (A) , (det A)s #(Ur,A )

+

a F (A)a F (A) ), (det A)s #(Ur,A

 m (O )r /Ur A∈Her

and Rr (s, F) =





| det(tr, p )|s−2l Kp

p

 m (O )r /Ur A∈Her

where Ur = G L m (K ) ∩ (G L m (C)tr



G L m (O p )tr−1 ),

p

Ur,A = {g ∈ Ur | g ∗ Ag = A}, = Ur,A ∩ S L m (K ). For an element F = (F1 , . . . , Fh )# ∈ Ur = Ur ∩ S L m (K ) and Ur,A −l (m) (m)  F) by S2l (U (Q)\U (A), det ) we also define the Rankin–Selberg series R(s,

 F) = R(s,

h  r =1

123

Rr∗ (s, Fr ).

Period of the adelic Ikeda lift for U (m, m)

77

We note that Rr∗ (s, Fr ) =

1 #(O∗ )

Rr (s, Fr ),

and hence  F) = R(s,

h 

1 #(O∗ )

Rr (s, Fr ).

r =1 (m)

Proposition 3.1 Let F be an element of S2l (Γi , det−l ). Put m L(i, χ i+1 ) 22lm+m−1 i=2 Rm = .   m−1 m D m(m−1)/2 i=0 L(2m − i, χ i ) i=1 C (i) C (2l − i + 1) Then Rr∗ (s, F) can be continued to a meromorphic function on the whole s-plane. Moreover it is holomorphic in s for Re(s) > 2l, and has a simple pole at s = 2l with the residue Rm F,F . O∗ Proof The assertion can be proved by a careful analysis of the proof of [[9], Proposition 22.2]. Here we have to treat more carefully than that because the group Γr is not necessarily a subgroup of Γ (m) , and we here give an outline of the proof. For a congruence subgroup Γ of Γ (m) and a character η of Γ, we define the non-holomorphic Siegel Eisenstein series E(Z , s; η, Γ ) by  E(Z , s; η, Γ ) = (det Y )s η(M)| j (M, Z )|−2s , 

M∈Γ∞ \Γ



A B ∈ Γ }. If η is the principal character we simply write it as 0 D E(Z , s; Γ ). In particular for a Dirichlet character η mod N we define the Eisenstein series (m) E(Z , s; η, Γ0 (N )) by  (m) η(M)| j (M, Z )|−2s . E(Z , s; η, Γ0 (N )) = (det Y ))s

where Γ∞ = {

(m)

M∈Γ0

(m)

(N )∞ \Γ0

(m)

For the F we can take a suitable N such that Γr

(2) m (s) = π m(m−1)/2

(N )

⊃ Γ (m) (N ). Put

m−1 

(s − ν),

ν=0 (2)

(2)

and  m,l (s) = m (s + 4l)(4π)−m(s+4l) . Then by Page 179 of [9], we have  s−2l p | det(ti, p )| K p (2)  m,l (s)Rr∗ (s, F) = #(O∗ )vol(Her m (C)/N Her m (O))  F(Z )F(Z )(det Y )2l E(Z , s¯ − 2l + m; Γ (m) (N ))d ∗ Z , × (m) (N )\Hm

where vol(Her m (C)/N Her m (O)) is the volume of Her m (C)/N Her m (O) with respect to the measure normalized so that vol(Her m (C)/Her m (O)) = 2m(1−m)/2 D m(m−1)/4 ,

123

78

H. Katsurada

and d ∗ Z = (det Y )−2m d X dY. By Lemma 17.2 of [9], we have  s−2l p | det(ti, p )| K p (2) ∗  m,l (s)Rr (s, F) = #(O∗ )vol(Her m (C)/N Her m (O))#X N   × F(Z )F(Z )(det Y )2l η∈X N

(m) (N )\Hm

(m)

E(Z , s¯ − 2l + m; η, Γ0

(N ))d ∗ Z ,

where X N is the set of Dirichlet characters mod N . It is well-known that E(Z , s − 2l + (m) m; η, Γ0 (N )) has a meromorphic continuation to the whole s-plane. Hence Rr∗ (s, F) can be continued to a meromorphic fucntion on the whole s-plane, and is holomorphic for Re(s) > (m) 2l. Moreover E(Z , s − 2l + m; η, Γ0 (N )) has a simple pole at s = 2l if and only if η is the trivial character η0 and by a careful analysis of the proof of [[9], Theorem 19.7] we have (m)

Ress=2l E(Z , s − 2l + m; η0 , Γ0

(N ))

(2) (m) = 2m(1−m)/2−1 |D|−m(m−1)/4 π m m  m i+1 −i i+1 )  ) i=2 L(i, χ i=2 (1 − p χ( p) × m−1 .  m−1 i −2n+i χ( p)i ) i=1 L(2m − i, χ ) p|N i=1 (1 − p m2

It is easily seen that vol(Her m (C)/N Her m (O)) = (N m 2m(1−m)/2 D −m(m−1)/4 )−1 , and #X N is the Euler function φ(N ). Hence Rr∗ (s, F) has a pole at s = 2l and 2

(2) m,l (s) = Ress=2l Rr∗ (s, F)

Rm F, FΓ (m) (N ) I N−1 #(O∗ )

(1)

for any 1 ≤ r ≤ h and N such that Γ (m) (N ) ⊂ Γr , where ⎞ ⎛   m−1 (1 − p −2n+i χ( p)i ) 2 ⎠. i=1 I N = ⎝φ(N )N m m −i i+1 ) i=2 (1 − p χ( p) p|N

We prove [Γ (m) : O∗ Γ (m) (N )] = I N .

(2)

To prove this, take a non-zero element G of S2l (Γ (m) ). Then (1) holds also for R1 (s, G) and we have Rm (2) G, GΓ (m) (N ) I N−1 m,l (s) = (3) Ress=2l R1∗ (s, G) #(O∗ ) On the other hand, since we have Γ (m) (1) ⊂ Γ (m) , again by (1), we have (2) m,l (s) = Ress=2l R1∗ (s, G)

Rm G, GΓ (m) (1) . #(O∗ )

(4)

Thus (1) can be proved by comparing (3) and (4) and remarking that [Γ (m) : Γ (m) (1)] = #(O∗ ). Hence Rr∗ (s, F) has a pole at s = 2l and (2)

m,l (s) = Ress=2l Rr∗ (s, F) (m)

for any 1 ≤ r ≤ h and F ∈ S2l (Γr

123

Rm F, F #(O∗ )

). This proves the assertion.

 

Period of the adelic Ikeda lift for U (m, m)

79

Remark The equality (2) can also be proved directly. Corollary (1) Let F be as in Proposition 3.1. Then Rr (s, F) can be continued to a meromorphic function on the whole s-plane. Moreover it is holomorphic in s for Re(s) > 2l, and has a simple pole at s = 2l with the residue Rm F, F.  F) (2) Let F = (F1 , . . . , Fh )# be an element of S2l (U (m) (Q)\U (m) (A), det −l ). Then R(s, can be continued to a meromorphic fucntion on the whole s-plane, and is holomorphic for  F) has a pole at s = 2l with the residue h ∗ Rm F, F. Re(s) > 2l. Moreover R(s, #O

4 Reduction to local computations  Li f t (m) ). To To prove our main result, we give explicit formulas for Rr (s, Im ( f )) and R(s, do this, we reduce the problem to local computations. Throughout the rest of this paper, let K p be a quadratic extension of Q p or K p = Q p ⊕ Q p . In the former case let O p be the ring of integers in K p , and f p the exponent of the conductor of K p /Q p , and put e p = f p − δ2, p , where δ2, p is Kronecker’s delta. In the latter case, put O p = Z p ⊕ Z p , and e p = f p = 0.  m (O p ). We note that Her  m (O p ) = p e p Her  m (O p ) = Her m (O p ) if K p is Moreover put Her not ramified over Q p . Let K be an imaginary quadratic extension of Q with the discriminant   = p|D p e p , and Her   m (O) = DHer −D. We then put D m (O ). Now let m and l be positive integers such that m ≥ l. Then for an integer a and Hermitian matrices A and B of degree m and l respectively with entries in O p put  Aa (A, B) = {X ∈ Mml (O p )/ p a Mml (O p ) | A[X ] − B ∈ p a H erl (O p )}, and Ba (A, B) = {X ∈ Aa (A, B) | rank O p / pO p X = l}.

Assume that A and B are non-degenerate. Then the number pa(−2ml+l ) #Aa (A, B) is independent of a if a is sufficiently large. Hence we define the local density α p (A, B) representing B by A as 2

α p (A, B) = lim pa(−2ml+l ) #Aa (A, B). 2

a→∞

In particular we write α p (A) = α p (A, A). Let {tr } be elements of G(A f ) defined as before.  For each T ∈ H er m (O p )× put F p(0) (T, X ) = F p ( p −e p T, X ) and p(0) (T, X ) = F p ( p −e p T, X ). F We remark that p(0) (T, X ) = X −ord p (det T ) X e p m− f p [m/2] F p(0) (T, p −m X 2 ). F For d ∈ Z× p . Put λm, p (d, X, Y ) =

  m (d,O p )/S L m (O p ) A∈Her

p(0) (A, X −1 ) F p(0) (A, Y −1 ) F . u p l p,A α p (A)

123

80

H. Katsurada

We note that λm, p (d, X −1 , Y −1 ) = λm, p (d, X, Y ). An explicit formula for λm, p ( pi d0 , X, Y ) will be given in the next section for d0 ∈ Z∗p and i ≥ 0. Theorem 4.1 Let f be a primitive form in S2k+1 (Γ0 (D), χ) or in S2k (S L 2 (Z)) according as m = 2n or 2n + 1. For such an f and a positive integer d0 put  λm, p (cr N K p /Q p (det tr, p )d0 , α p , α p ), am,r ( f ; d0 ) = p

where cr =

 p

p −ord p (det tr, p det tr, p ) , and α p is the Satake p-parameter of f. Moreover put μm,k,D = D m(s−2k+l0 )+(2k−l0 )[m/2]−m(m+1)/4−1/2 m  × 2−c D m(s−2k−2n)−m+1 C (i), i=2

where l0 = 0 or 1 according as m is even or odd. Then for Re(s) >> 0, we have Rr (s, Im ( f )r ) = μm,k,D

∞ 

am,r ( f ; d0 )d0−s+2k+2n .

d0 =1

Proof The assertion can be proved by using the same argument as in the proof of [[[5]], Theorem 4.1].    For a G L m (O p )-invariant function ω p on H er m (O p )× put 

m, p (ω p , X, Y, t) := H

 m (O p )/G L m (O p ) A∈Her

ω p (A)t ord p (det A)

(0) (0) F˜ p (A; X ) F˜ p (A; Y ) . α p (A)

We note that F˜ p(0) (A; X ) F˜ p(0) (A; Y ) = F˜ p(0) (A; X −1 ) F˜ p(0) (A; Y −1 ). ×  er mp taking the value 1, and εm, p the function of Let ιm, p be the constant function of H  H er m (O p )× defined by  1 if det A ∈ N K p /Q p (K p ) εm, p (A) = . −1 otherwise

We sometimes drop the suffix and write ιm, p as ι and the others if there is no fear of confusion. m, p (ω p ; X, Y, t) will be given in the next section for ω p = ιm, p An explicit formula for H and εm, p . Theorem 4.2 Let f be a primitive form in S2k+1 (Γ0 (D), χ) or in S2k (S L 2 (Z)) according as m = 2n or 2n + 1. For such an f let Li f t (m) ( f ) be the lift of f to S2k+2n (U (m) (Q)\U (m) (A), det −k−n ). Then for Re(s) >> 0, we have  1/2 (1)  D μ m,k,D C (m)  Li f t ( f )) = m, p (ι p ; α p , α p , p −s+2k+2n ) R(s, H × 2 p   −s+2k+2n  + ) , Hm, p (ε p ; α p , α p , p p

where α p is the Satake p-parameter of f.

123

Period of the adelic Ikeda lift for U (m, m)

81

Proof The assertion can be proved by using the same argument as in the proof of [[3], Theorem 3.4].  

5 Proof of Theorem 2.1 Theorem 5.1 Let k and n be positive integers. Let f be a primitive form in S2k+1 (Γ0 (D), χ). For a subset Q of Q D and a Dirichlet character η = χ i−1 with a positive integer i put ⎧ ⎨ i −s (1 − α 2p χ i ( p)χ Q ( p) p −s )(1 − α −2 M(s, f, Ad, η, χ Q ) = p χ ( p)χ Q ( p) p ) ⎩ p ∈Q /  (1 − α 2p χ Q ( p)χ i−1 ( p) p −s ) × (1 − χ i−1 ( p)χ Q ( p) p −s )2 p∈Q i−1 ( p) p −s )(1 − χ Q ( p)χ i ( p) p −s )2 × (1 − α −2 p χ Q ( p)χ

⎫−1 ⎬ ⎭

,

we make the convention η( p)χ j ( p) = where for a Dirichlet character ψ = χ Q or η = χ Q η( p) or 0 according as j is even or odd. Then, we have

Rr (s, I2n ( f )r ) = D ns+n ×

2n 

2 −n/2−1/2

(i, χ i ) 

i=2

2n−1 

L(2s − 4k − i, χ i )−1

i=0



×

2−2n+1

χ Q ((−1)n cr )

Q⊂Q D

2n 

M(s − 2k − 2n + i, f, Ad, χ i−1 , χ Q ).

i=1

Proof The assertion can be proved by Theorem 4.1 and [[5], Theorems 5.5.2 and 5.5.4] using the same argument as in pages 155–156 of [3].   Corollary Rr (s, I2n ( f )r ) = D ns+n ×

2n 

2 −n/2−1/2

(i, χ i ) 

i=2

×

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

ηn,Q ( f )

2−2n+1

2n−1 

L(2s − 4k − i, χ i )−1

i=0 2n 

L(s − 2k − 2n + i, f, Ad, χ i−1 )L(s −2k − 2n+i, χ i−1 )

i=1

⎫ ⎪ ⎪ ⎬  n i−1 χ Q ((−1) cr ) M(s − 2k − 2n + i, f, Ad, χ , χ Q ) . + ⎪ ⎪ Q⊂Q D i=1 ⎭ 2n 

f Q = f

Proof The assertion can be proved by [[5], Lemma 6.3].

 

123

82

H. Katsurada

Theorem 5.2 Let k and n be positive integers. Given a primitive form f ∈ S2k (S L 2 (Z)). Then, we have Rr (s, I2n+1 ( f )r ) = D ns+n ×

2n+1  i=2

×

2n+1 

2 +3n/2+1/2

(i, χ i ) 

2−2n

2n 

L(2s − 4k − i + 2, χ i )−1

i=0

L(s − 2k − 2n + i, f, Ad, χ i−1 )L(s − 2k − 2n + i, χ i−1 ).

i=1

Proof The assertion follows from Theorems 4.1 and [[5], Theorem 5.5.3].

 

Proof of Theorem 2.1 The assertion (1) can be proved by Corollary to Theorem 5.1 and [[5], Proposition 6.4] similarly to [[5], Theorem 2.1, (1).] Similarly, the assertion (2) can be proved by Theorem 5.2 and [[5], Proposition 6.4].  

6 Explicit formulas of formal power series of Rankin–Selberg type To give another proof to Theorem 2.2, in this section, we give an explicit formula of m, p (ω, X, Y, t). To do this, we define another formal power series. Throughout this section H ! ! for a G L r (O p )-stable subset B of Her m (Q p ), we simply write T ∈B instead of T ∈B/∼ if there is no fear of confusion. We also simply write ord p as ord and the others if the prime number p is clear from the context. We also write ν K p as ν. For d ∈ Z p we put Her m (O p ; d) = Her m (O p ; d) ∩ Her m (O p ). for and put λ∗m, p (d, X, Y ) =

  m (d N K p /Q p (O∗p ),O p )/G L m (O p ) A∈Her

p(0) (A, Y ) p(0) (A, X ) F F . α p (A)

We then define Hˆ m, p (d0 , X, Y, t) =

∞ 

λ∗m, p ( pi d0 , X, Y )t i .

i=0

It is well-known that #(Z∗p /N K p /Q p (O∗p )) = 2 if K p /Q p is ramified. Hence we can take a complete set N p of representatives of Z∗p /N K p /Q p (O∗p ) so that N p = {1, ξ0 } with χ K p (ξ0 ) = −1. Here χ K p (b) = (−D, b) p for b ∈ Z p , b  = 0. As for the relation between these two formal power series, we easily obtain: Proposition 6.1 (1) Assume that K p is an unramified quadratic extension of Q p . Then we have m, p (ι, X, Y, t) = Hˆ m, p (d0 , X, Y, t), H and m, p (ε, X, Y, t) = Hˆ m, p (d0 , X, Y, −t) H for any d0 ∈ Z∗p .

123

Period of the adelic Ikeda lift for U (m, m)

83

(2) Assume that K p = Q p ⊕ Q p . Then we have m, p (ι, X, Y, t) = Hˆ m, p (ε, X, Y, t) = Hˆ m, p (d0 , X, Y, t), H for any d0 ∈ Z∗p . (3) Assume that K p is ramified over Q p . Then we have m, p (ι, X, Y, t) = H



Hˆ m, p (d, X, Y, t),

d∈N p

and m, p (ε, X, Y, t) = H



χ K p (d) Hˆ m, p (d, X, Y, t).

d∈N p

m (1 − q i ). Put φm (q) = i=1 An explicit formula for Hˆ m, p (d, X, Y, t) has been given in [5]. Hence we have Theorem 6.2 Let m = 2n be even. (1) Assume that K p is an unramified quadratic extension of Q p . Then 2n (ι, X, Y, t) = H

2n

−2n−i t 2 ) i=1 (1 − (− p) φ2n (− p −1 )

1

× 2n

−2n+i−1 X Y t)(1 − (− p)−2n+i−1 X Y −1 t) i=1 (1 + (− p) 1 × 2n , −2n+i−1 X −1 Y t)(1 + (− p)−2n+i−1 X −1 Y −1 t) (1 − (− p) i=1

and 2n (ι, X, Y, t) = H

2n

−2n−i t 2 ) i=1 (1 + (− p) −1 φ2n (− p )

1

× 2n i=1

(1 − (− p)−2n+i−1 X Y t)(1 + (− p)−2n+i−1 X Y −1 t) 1

× 2n

−2n+i−1 X −1 Y t)(1 − (− p)−2n+i−1 X −1 Y −1 t) i=1 (1 + (− p)

.

(2) Assume that K p = Q p ⊕ Q p . Then 2n (ω, X, Y, t) = H

2n

p −2n−i t 2 ) φ2n ( p −1 )

i=1 (1 −

× 2n

1

−2n+i−1 X Y t)(1 − p −2n+i−1 X Y −1 t) i=1 (1 − p 1 × 2n −2n+i−1 X −1 Y t)(1 − p −2n+i−1 X −1 Y −1 t) (1 − p i=1

for ω = ι, ε.

123

84

H. Katsurada

(3) Assume that K p is a ramified quadratic extension of Q p . Then 2n (ι, X, Y, t) = t ni p H

n

p −2n−2i t 2 ) φn ( p −2 )

i=1 (1 −

1

× n

i=1 (1 −

p −2n−2i−1 X Y t)(1 −

p −2n−2i−1 X −1 Y −1 t)

,

and n

p −2n−2i t 2 ) φn ( p −2 ) 1 . × n −2n−2i −1 X Y (χ K p ( p)t))(1 − p −2n−2i X Y −1 (χ K p ( p)t)) i=1 (1 − p

2n (ε, X, Y, t) = χ K p ((−1)n )(χ K p ( p)t)ni p H

i=1 (1 −

Theorem 6.3 Let m = 2n + 1 be odd, and ω = ι or ε. (1) Assume that K p is unramified over Q p . Then 2n+1 (ω, X, Y, t) = H

2n+1 i=1

(1 − (− p)−2n−i−1 t 2 ) φ2n+1 (− p −1 )

× 2n+1 i=1

× 2n i=1

1 (1 + (− p)−2n+i−2 X Y t)(1 + (− p)−2n+i−1 X Y −1 t) 1

(1 + (− p)−2n+i−2 X −1 Y t)(1 + (− p)−2n+i−2 X −1 Y −1 t)

(2) Assume that K p = Q p ⊕ Q p . Then 2n+1 (ω, X, Y, t) = H

2n+1 i=1

(1 − p −2n−i−1 t 2 ) φ2n+1 ( p −1 )

× 2n+1 i=1

× 2n+1 i=1

1 (1 −

p −2n+i−2 X Y t)(1 −

p −2n+i−2 X Y −1 t)

(1 −

p −2n+i−2 X −1 Y t)(1 −

1 p −2n+i−2 X −1 Y −1 t)

(3) Assume that K p is ramified over Q p . Then 2n+1 (ι, X, Y, t) = t (n+1)i p +δ2 p H × n+1

n+1

p −2n−2i t 2 ) φn ( p −2 ) 1

i=1 (1 −

p −2n+2i−3 X Y t)(1 − p −2n+2i−3 X −1 Y −1 t) 1 × , (1 − p −2n+2i−3 X −1 Y t)(1 − p −2n+2i−3 X Y −1 t) i=1 (1 −

2n+1 (ε, X, Y, t) = 0. and H

123

.

.

Period of the adelic Ikeda lift for U (m, m)

85

7 Another proof of Theorem 2.2 Theorem 7.1 Let k and n be positive integers. Given a primitive form f ∈ S2k+1 (Γ0 (D), χ D ). Then, we have  Li f t (2n) ( f )) = D ns+n 2 −n/2 2−2n R(s, ×

 2n 

2n 

(i, χ i ) 

2n−1 

i=1

L(2s − 4k − i, χ i )−1

i=0

L(s − 2k − 2n + 2i − 1, f, Ad, χ i−1 )

i=1

× L(s − 2k − 2n + 2i − 1, χ i−1 ) + χ((−1)n )

2n 

L(s − 2k − 2n + 2i − 1, f, Ad, χ i )

i=1

"

× L(s − 2k − 2n + 2i − 1, χ ) . i

 

Proof The assertion follows directly from Theorems 4.2 and 6.2. Corollary (1) Assume that f Q D = f. Then  Li f t (2n) ( f )) = D ns+n 2 −n/2 2−2n R(s,

2n 

(i, χ i ) 

i=1

× ηn ( f )

2n 

2n−1 

L(2s − 4k − i, χ i )−1

i=0

L(s − 2k − 2n + 2i − 1, f, Ad, χ i−1 )

i=1

× L(s − 2k − 2n + 2i − 1, χ i−1 ). (2) Assume that f Q D  = f. Then  Li f t (2n) ( f )) = D ns+n 2 −n/2 2−2n R(s, ×

 2n 

2n 

(i, χ i ) 

i=1

2n−1 

L(2s − 4k − i, χ i )−1

i=0

L(s − 2k − 2n + 2i − 1, f, Ad, χ i−1 )

i=1

× L(s − 2k − 2n + 2i − 1, χ i−1 ) + χ((−1)n )

2n 

L(s − 2k − 2n + 2i − 1, f, Ad, χ i )

i=1

"

×L(s − 2k − 2n + 2i − 1, χ ) , i

2n L(s − 2k − 2n + 2i − 1, f, Ad, χ i )L(s − 2k − 2n + 2i − 1, χ i ) is holomorphic and i=1 at s = 2k + 2n. Proof The assertion follows from [[5], Lemma 6.3, Proposition 6.4.]

 

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H. Katsurada

Theorem 7.2 Let k and n be positive integers. Given a primitive form f ∈ S2k (S L 2 (Z)). Then, we have  Li f t (2n+1) ( f )) = D ns+n 2 +3n/2+1 2−2n−1 R(s,

2n+1 

(i, χ i ) 

i=1

×

2n+1 

2n 

L(2s − 4k − i, χ i )−1

i=1

L(2s − 2k + 2i −1, f, Ad, χ i−1 )L(2s − 2k +2i − 1, χ i−1 ).

i=1

Proof The assertion follows directly from Theorems 4.2 and 6.3.

 

Proof of Theorem 2.2 We note that (1, χ). h K = 2−1 D 1/2 #(O∗ ) Thus the assertion (1) follows from (2) of Corollary to Proposition 3.1 and Corollary to Theorem 7.1, and the assertion (2) follows from (2) of Corollary to Proposition 3.1 and Theorem 7.2.   Correction of [3] p.141, l. 5↑: For ’[R Omp + Omp : Omp ]’, read ’[R Omp + Omp : Omp ]1/2 ’. p. 147, l. 19: For ’Res K /Q (Her m )’, read ’Her m ’. p.147, l. 1↑: For ’(1+ p −1 )−1 ’ and ’(1− p −1 )−1 ’, read ’1+ p −1 ’ and ’1− p −1 ’,respectively. p. 148, l. 1: For ’{ u −1 read ’{ u p }’. p }’,  −1 u p ’, read ’ p  u p ’. p. 149, ls. 5 and 6: For ’ p  p154, l. 2: Insert ‘D −1/2 C (1)−1 after μm,k,D .

References 1. Brown, J., Keaton, R.: Congruence primes for Ikeda lifts and the Ikeda ideal. Pac. J. 274, 27–52 (2015) 2. Ikeda, T.: On the lifting of hermitian modular forms. Compos. Math. 1144, 1107–1154 (2008) 3. Katsurada, H.: Koecher-Maaß series of the adelic Hermitian Eisenstein series and the adelic Hermitian Ikeda lift for U (m, m). Comment. Math. Univ. St. Pauli 63, 137–159 (2014) 4. Katsurada, H.: Congruence between Duke-Imamoglu-Ikeda lifts and non-Duke-Imamoglu-Ikeda lifts. Comment. Math. Univ. St. Pauli 64, 109–129 (2015) 5. Katsurada, H.: On the period of the Ikeda lift for U (m, m), To appear in Math. Z. Math. arXiv:1102.4393 [math. NT] 6. Katsurada, H., Kawamura, H.: On Ikeda’s conjecture on the period of the Duke-Imamoglu-Ikeda lift. Proc. Lond. Math. Soc. 111, 445–483 (2015) 7. Klosin, K.: The Maass space for U (2, 2) and the Bloch–Kato conjecture for the symmetric square motive of a modular form. J. Math. Soc. Jpn. 67, 797–859 (2015) 8. Shimura, G.: Euler Products and Eisenstein Series. CBMS Reginal Conference Series in Mathmatics, vol. 93. American Mathmatical Society, Providence (1997) 9. Shimura, G.: Arithmeticity in the Theory of Automorphic Forms. Mathematical Surveys and Monographs, vol. 82. American Mathmatical Society, Providence (2000)

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