On the Yoshida lift and the Doi-Naganuma lift.
Takeo Okazaki Osaka University
Main Results.
Let q be 2 or a prime number congruent to 1 modulo 4. PutK=Q(√q) and let ψ =ψK be the quadratic character of Q related to the extension of K/Q. Letρ= detk×Sym(2v) be a polynomial representation ofGLn(C). We denote by Mρn(Γ, ψ) the space of vector valued holomorphic Siegel modular form F of degree n such that
F(γ(Z)) =ψ(det(d)) det(j(γ, Z))k×ρ0(cZ+d)F(Z), Z ∈Hn of type ρ for a congruence subgroup Γ, and bySρn(Γ, ψ) that of cuspforms.
Then our results are as follows.
Theorem 1 Suppose that f ∈S2v+21 (Γ0(q), ψ)is an eigenform. There exists a non-cuspform Eh such that
1) Ef ∈Mdet2 2×Sym(2v)(Γ0(q), ψ).
2) Its spinor L-function Z(s, Ef) is equal to L(s, f)L(s, f), up to the Euler q-factor.
3) For the Siegel operatorΦ, we haveEf|Φ =f+f, up to constant multiple.
4) In case of v = 0, the field Q(Ef) generated by the Fourier coefficients of Ef are contained in Q(f).
In case of v = 0, the phenomenon 4) can be considered as a Klingen- Eisenstein lift of Nebentypus, outside of the range of convergence. We note that the relationQ(Ef)⊂Q(f) holds in such a range (c.f. [4]). The theorem is an analogy of the Doi-Naganuma lift of Nebentypus.
Further, our second result is the next.
Theorem 2 Suppose that f ∈ S2v+21 (Γ0(2e), ψ) with e ∈ Z>0 is obtained from a Gr¨oßen-character on Q(√
−2) or Q(√
−1). Then, there exists a pair of a Siegel cuspform FC and non-cuspform Ef such that
Z(s, FC) =Z(s, Ef) =L(s, f)L(s, f), up to finitely many primes.
However, we remark that FC has the central characterψK andEf has trivial one. So, their Satake parameters don’t coincide. Their standardL-functions are not the same one.
1 Liftings.
The above Siegel modular forms Ef and FC are obtained by the Yoshida lift, and B¨ocherer and Schulze-Pillot’s refinements of Yoshida lift for vector valued case. There are two types of the Yoshida lift. First type (we label Y(I)) provides a Siegel modular form Yf1×f2 whose spinor L-function is, up to finitely many bad primes, equal to a products of L-function of elliptic cuspforms f1 and f2:
Z(s, Yf1×f2) =L(s, f1)L(s, f2). (1.1) Second type (labeled (Y(II)) provides a Siegel modular form whose spinorL- function is, up to finitely many bad primes, equal to L-function of a Hilbert cuspform g over a real quadratic field:
Z(s, Yg) =L(s, g). (1.2)
Strictly speaking, in these constructions Yoshida lift, we need to give auto- morphic forms on definite quaternion algebras having same L-functions offi andg. In our situations, Jacquet-Langlands theory assures that it is possible.
The Doi-Naganuma lift of neben-typus provides a Hilbert cuspform fDN over a real quadratic field K, whose L-function is equal to a products of L-function of elliptic cuspforms f and its complex conjugate f:
L(s, fDN) =L(s, f)L(s, f). (1.3) As to Theorem 1, the non-cuspform Ef is obtained from a Hilbert cusp- formg =fDN by the Yoshida lift of second type. However, we need to check Ef 6= 0. We see it later.
Further, suppose that f ∈ S2v+2(Γ0(2e), ψK) is obtained from a Gr¨oßen- character µon Q(√
−2) or Q(√
−1):
f(z) =θµ(z) =X
µ(a) exp(2πiz),
where aruns all over integral ideals prime to 2. Then, we obtain a cuspform Yf×f by the Yoshida lift of first type. And we obtain a non-cuspform from Hilbert cuspform g =fDN by the Yoshida lift of second type. Summing up (1.1), (1.2), and (1.3), we get Theorem 2.
Sdet2×Sym(2v)(Sp2(Q))3Yf×f =FC Ef =YfDN ∈Mdet2×Sym(2v)(Sp2(Q))
S2v+2(Γ , ψK)3f, f fDN ∈S2v+2(ΓK)
6 6
?
Y(I) Y(II) Φ
DN -
(In what relation?)
DN∨
-
2 Non-vanishing.
Following the idea of B¨ocherer and Schulze-Pillot [2], we see the non-vanishing of Ef = YfDN. Since, the Yoshida lift is a theta lift, YfDN = Yf(2)DN can be seen as the image of Yf(3)DN by Siegel operator Φ.
Kitaoka [5]’s result saysYf(3)DN 6= 0. So, in order to seeYf(2)DN 6= 0, we show that Yf(3)DN 6∈ Sρ(Γ0(q)) by concluding a contradiction under the assumption Yf(3)DN ∈Sρ(Γ0(q)). There is a pullback formula such as
< G2nρ (Z, W;s), F(Z)>= νρ0(s+k)L(ˇq)(2s+k−n, F, ψ)F\(W) L(2s+k, ψ)Qn
i=1ζ(ˇq)(4s+ 2k−2i) (2.4) for any holomorphic cuspform F(Z)∈Sρn(Γ0(q)). Here Gρ is obtained from Garrett’s Eisenstein series [3] by a differenetial operator. νρ0(s) is a certain meromorphic function of s and has at least one pole at s= 3.
L(ˇq)(2s+k−n, F) is the standard L-function of F outside of q-factor and L(ˇq)(2s+k−n, F, ψ) is its ψ-twist. F\ has the complex conjugate Fourier coefficients of F.
Then comparing the number of poles at s = 1 of both sides of (2.4), we are going to obtain the contradiction. Kudla and Rallis [6] says the left side has at most a simple pole at s = 1. However, L(ˇq)(s, Yf(3)DN) = G(ˇq)(s, fDN) and Asai [1] says G(ˇq)(s, fDN) has a pole ats= 1. So, the right side of (2.4) has double poles at s= 1.
By the way the non-vanishing of Yf×f is shown by observing a certain period of f ×f is not zero.
3 Example.
We give the example of Theorem 2. The Jacobian J(C) of the hyper-elliptic curve C defined by y2 = x5 − x has a complex multiplication (x, y) → (ix,exp(2πi8 )y). So, End(J(C)) ⊃ L = Q(exp(2πi8 )). Since Gal(L/Q) ' (Z/2Z)2, J(C) is not absolutely simple, but Q-simple. Indeed, J(C) is Q- isogenous to a Weil restriction ResM/ E of a certain elliptic curve E defined over a quadratic field M with j(E) ∈ Q. Here E is given by the equation y2 =x(x−1)(x−(1 +√
2)) and has a complex multiplication of Q(√
−2).
Due to Shimura’s CM-theory, the Hasse-Weil zeta function of J(C) is written by a Gr¨oßen-character λ on L in the form λ = µ◦NL/ (√−2) for a certain µ on Q(√
−2). This µ has the property described as in Theorem 2 and the conductor of µis 2√
−2. (From this we know the conductor of J(C) is 212.)
Then, there exist non-cuspformEf ∈M2(Γ0(64), ψ) and cuspformFC ∈ S2(Γ(8)) whose spinor L-functions are equal to the Hasse-Weil zeta function of J(C).
References
[1] T. Asai: On Certain Dirichlet Series Associated with Hilbert Modular Forms and Rankin’s Method, Math. Ann. 226, (1977) 81-94.
[2] S. B¨ocherer and R. Schulze-Pillot: Siegel modular forms and theta series attached to quaternion algebras II, Nagoya Math. J.147(1997), 71-106.
[3] P. Garrett: On the arithmetic of Siegel-Hilbert Modular cuspforms: Pe- tersson inner products and Fourier coefficients, Invent. Math.107(1992) 453-481.
[4] M. Harris: Eisenstein series and Shimura varieties, Annals of Math,119 (1984), 59-94.
[5] Y. Kitaoka: Representations of quadratic forms and their applications to Selberg’s zeta function, Nagoya Math J, 63 (1976), 153-162.
[6] S. Kudla and S. Rallis: A regularized Siegel-Weil formula: The first term identity, Annals of Math, 140 (1994), 1-80.
[7] G. Shimura: Abelian varieties with complex multiplication and modular functions, Princeton univ press, (1998).
[8] H. Yoshida: Siegel’s modular forms and the arithmetics of quadratic forms, Invent. math. 60 (1980), 193-248.
[9] —: On Siegel modular forms obtained by theta series, J. Reine. Angrew.
Math, 352 (1984), 184-219.
Department of Mathematics Graduate School of Science Osaka University, Machikaneyama 1-16, Toyonaka, Osaka, 560-0043, Japan
E-mail: [email protected]
