ATSUSHI ICHINO
In this note, we announce that Ikeda’s conjecture [12] holds forr = 1 and n = 0.
1. Statement of the main theorem Letκ be an odd positive integer. Let
f(τ) = X
N >0
af(N)qN ∈S2κ(SL2(Z))
be a normalized Hecke eigenform and
h(τ) = X
N >0
−N≡0,1 mod 4
ch(N)qN ∈Sκ+1/2+ (Γ0(4))
a Hecke eigenform associated tof by the Shimura correspondence. Let F(Z) = X
B>0
A(B)e2π√−1 tr(BZ) ∈Sκ+1(Sp2(Z))
be the Saito-Kurokawa lift of h, where A
µµ n r/2 r/2 m
¶¶
= X
d|(n,r,m)
dκch
µ4nm−r2 d2
¶ .
For each normalized Hecke eigenform g(τ) = X
N >0
ag(N)qN ∈Sκ+1(SL2(Z)),
we consider the period integral hF|h1×h1, g×gigiven by hF|h1×h1, g×gi
= Z
SL2(Z)\h1
Z
SL2(Z)\h1
F
µµτ1 0 0 τ2
¶¶
g(τ1)g(τ2)y1κ−1yκ−12 dτ1dτ2.
1
Define the Petersson norms off, g,h by hf, fi=
Z
SL2(Z)\h1
|f(τ)|2y2κ−2dτ, hg, gi=
Z
SL2(Z)\h1
|g(τ)|2yκ−1dτ, hh, hi= 1
6 Z
Γ0(4)\h1
|h(τ)|2yκ−3/2dτ, respectively.
For each prime p, let {αp, α−1p } and {βp, βp−1} denote the Satake parameters of g and f at p, respectively. Then
1−ag(p)X+pκX2 = (1−pκ/2αpX)(1−pκ/2α−1p X), 1−af(p)X+p2κ−1X2 = (1−pκ−1/2βpX)(1−pκ−1/2βp−1X).
We put
Ap =pκ
α2p 0 0
0 1 0
0 0 α−2p
, Bp =pκ−1/2
µβp 0 0 βp−1
¶ .
Define the L-functionL(s,Sym2(g)⊗f) by an Euler product L(s,Sym2(g)⊗f) =Y
p
det(16−Ap⊗Bp·p−s)−1
for Re(s) À 0. Let Λ(s,Sym2(g) ⊗f) be the completed L-function given by
Λ(s,Sym2(g)⊗f) = ΓC(s)ΓC(s−κ)ΓC(s−2κ+ 1)L(s,Sym2(g)⊗f), where ΓC(s) = 2(2π)−sΓ(s). It satisfies the functional equation
Λ(4κ−s,Sym2(g)⊗f) = Λ(s,Sym2(g)⊗f).
Our main result is as follows.
Theorem 1.1.
Λ(2κ,Sym2(g)⊗f) = 2κ+1hf, fi hh, hi
|hF|h1×h1, g×gi|2 hg, gi2 . Theorem 1.1 has an application to Deligne’s conjecture [3].
Corollary 1.2. For σ ∈Aut(C), µΛ(2κ,Sym2(g)⊗f)
hg, gi2c+(f)
¶σ
= Λ(2κ,Sym2(gσ)⊗fσ) hgσ, gσi2c+(fσ) . Here c+(f) is the period off as in [19].
Proof. The assertion follows from Theorem 1.1 and the Kohnen-Zagier formula [15]
Λ(κ, f, χ−D) = 2−κ+1D1/2|ch(D)|2hf, fi hh, hi,
where −D<0 is a fundamental discriminant. ¤ Remark 1.3. It seems that Corollary 1.2 does not follow from the alge- braicity of central critical values of triple product L-functions. Notice that
Λ(2κ, g⊗g⊗f) = Λ(2κ,Sym2(g)⊗f)Λ(κ, f) = 0.
2. Proof of Theorem 1.1
Since F is a cusp form, the usual unfolding method does not work.
Instead, we use seesaws in the sense of Kudla [16].
We may assume that ch(N) ∈R for all N ∈N. Let f and g denote the automorphic forms on GL2(AQ) associated tof andg, respectively.
Let h and Θ denote the automorphic forms on SL^2(AQ) associated to handθ, respectively. Hereθ(τ) = P
N∈ZqN2 is the theta function. Let π be the irreducible cuspidal automorphic representation of GL2(AQ) generated byg.
Proposition 2.1. For the seesaw O(3,2)
OO OO OO OO OO OO
OO SL2×SLf2
oooooooooooo
O(2,2)×O(1) SLf2 ,
the identity
hF|h1×h1, g×gi= 2κ+2ξ(2)hg, gihhΘ,g]i holds. Here g] ∈π and ξ(s) =π−s/2Γ(s/2)ζ(s).
Fix a fundamental discriminant −D < 0 with −D ≡ 1 mod 8 such that Λ(κ, f, χ−D) 6= 0 (and hence ch(D) 6= 0). Such a discriminant exists by [21], [2]. Let πK be the base change of π to the imaginary quadratic field K=Q(√
−D).
Proposition 2.2. For the seesaw SLf2×SLf2
PP PP PP PP PP
PP O(3,1)
nnnnnnnnnnnnnn
SL2 O(2,1)×O(1) ,
the identity
hhΘ,g]i= (√
−1)κD−1/2ch(D)−1hf, fi−1hh, hiI(g]K,f) holds. Here g]K ∈πK and
I(g]K,f) = Z
A×QGL2(Q)\GL2(AQ)
g]K(h)f(h)dh.
The following proposition follows from the regularized Siegel-Weil formula by Kudla and Rallis [17] and the integral representation of triple productL-functions by Garrett [5], Piatetski-Shapiro and Rallis [18].
Proposition 2.3. For the seesaw Sp3
TT TT TT TT TT TT TT TT TT
TT RK/QO(2,2)×O(2,2)
jjjjjjjjjjjjjjj
RK/QSL2×SL2 O(2,2)
,
the identity
Λ(2κ,Sym2(g)⊗f)Λ(κ, f, χ−D) = −22κ+6D−1/2ξ(2)2I(gK],f)2 holds.
Now Theorem 1.1 follows from Propositions 2.1–2.3 and the Kohnen- Zagier formula [15].
3. The Gross-Prasad conjecture
In this section, we interpret our result in terms of the Gross-Prasad conjecture [6], [7], which has been refined in a joint work with Tamotsu Ikeda [11].
Let H1 = SO(n+ 1) and H0 = SO(n) be special orthogonal groups over a number field k with embedding ι : H0 ,→ H1. Let πi ' ⊗vπi,v be an irreducible cuspidal automorphic representation of Hi(Ak). We assume that
HomH0(kv)(π1,v, π0,v)6= 0 for all v.
Conjecture 3.1(Gross-Prasad). Assume thatπ1 andπ0 are tempered.
Then the period integral hF1|H0, F0i=
Z
H0(k)\H0(Ak)
F1(ι(h0))F0(h0)dh0
does not vanish for some F1 ∈π1 and some F0 ∈π0 if and only if L
µ1
2, π1×π0
¶ 6= 0.
To relate our result to the Gross-Prasad conjecture, we would like to remove the assumption that π1 and π0 are tempered, and give an explicit formula for the period integral in terms of special values. We put
Pπ1,π0(s) = L(s, π1×π0) L¡
s+ 12, π1,Ad¢ L¡
s+ 12, π0,Ad¢,
where Ad :LHi →GL(Lie(LHi)) is the adjoint representation. In [11], we conjectured that the identity
|hF1|H0, F0i|2
hF1, F1ihF0, F0i =Pπ1,π0 µ1
2
¶
holds up to an elementary constant, and gave an example for n = 5 with non-temperedπ1,π0. Also, this conjectural identity is compatible with the results of Waldspurger [20] for n = 2, Harris and Kudla [8], [9] for n = 3, B¨ocherer, Furusawa, and Schulze-Pillot [1] forn = 4.
We now discuss the case n = 4. Letπ1 (resp. π0) be the irreducible cuspidal automorphic representation of SO(3,2)(AQ) ' PGSp2(AQ) (resp. SO(2,2)(AQ) ' [GL2(AQ) × GL2(AQ)]0/A×Q) generated by F (resp. g×g). Let π (resp. σ) be the irreducible cuspidal automorphic representation of GL2(AQ) generated by g (resp. f). Then
L(s, π1) = L(s, σ)ζ µ
s+1 2
¶ ζ
µ s− 1
2
¶ , L(s, π0) = L(s, π×π).
By Theorem 1.1 and the result of Kohnen and Skoruppa [14],
|hF|h1×h1, g×gi|2
hF, Fihg, gi2 =Pπ1,π0 µ1
2
¶ .
This identity might hold even if F is not a Saito-Kurokawa lift. Using Dokchitser’s computer program [4] and Katsurada’s formula [13] for hF, Fi, one might check it numerically.
References
[1] S. B¨ocherer, M. Furusawa, and R. Schulze-Pillot, On the global Gross-Prasad conjecture for Yoshida liftings, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, 2004, pp. 105–130.
[2] D. Bump, S. Friedberg, and J. Hoffstein, Nonvanishing theorems for L- functions of modular forms and their derivatives, Invent. Math. 102 (1990), 543–618.
[3] P. Deligne,Valeurs de fonctionsLet p´eriodes d’int´egrales, Automorphic forms, representations andL-functions, Proc. Sympos. Pure Math.33, Amer. Math.
Soc., 1979, pp. 313–346.
[4] T. Dokchitser, Computing special values of motivic L-functions, Experiment.
Math.13(2004), 137–149.
[5] P. B. Garrett,Decomposition of Eisenstein series: Rankin triple products, Ann.
of Math.125(1987), 209–235.
[6] B. H. Gross and D. Prasad, On the decomposition of a representation ofSOn
when restricted toSOn−1, Canad. J. Math.44 (1992), 974–1002.
[7] , On irreducible representations ofSO2n+1×SO2m, Canad. J. Math.
46(1994), 930–950.
[8] M. Harris and S. S. Kudla, The central critical value of a triple product L- function, Ann. of Math.133(1991), 605–672.
[9] ,On a conjecture of Jacquet, Contributions to automorphic forms, ge- ometry, and number theory, Johns Hopkins Univ. Press, 2004, pp. 355–371.
[10] A. Ichino,Pullbacks of Saito-Kurokawa lifts, preprint.
[11] A. Ichino and T. Ikeda,On Maass lifts and the central critical values of triple productL-functions, preprint.
[12] T. Ikeda, Pullback of the lifting of elliptic cusp forms and Miyawaki’s conjec- ture, preprint.
[13] H. Katsurada,Special values of the standard zeta function of a Hecke eigenform of degree2, preprint.
[14] W. Kohnen and N.-P. Skoruppa,A certain Dirichlet series attached to Siegel modular forms of degree two, Invent. Math.95(1989), 541–558.
[15] W. Kohnen and D. Zagier,Values of L-series of modular forms at the center of the critical strip, Invent. Math.64(1981), 175–198.
[16] S. S. Kudla, Seesaw dual reductive pairs, Automorphic forms of several vari- ables (Katata, 1983), Progr. Math.46, Birkh¨auser Boston, 1984, pp. 244–268.
[17] S. S. Kudla and S. Rallis, A regularized Siegel-Weil formula: the first term identity, Ann. of Math.140(1994), 1–80.
[18] I. I. Piatetski-Shapiro and S. Rallis, Rankin triple L functions, Compositio Math.64(1987), 31–115.
[19] G. Shimura, On the periods of modular forms, Math. Ann.229 (1977), 211–
221.
[20] J.-L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leur centre de sym´etrie, Compositio Math.54(1985), 173–242.
[21] , Correspondances de Shimura et quaternions, Forum Math.3(1991), 219–307.
Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
E-mail address: [email protected]