SIEGEL CUSP FORMS
TAMOTSU IKEDA
§ History
In late 70’s, Saito and Kurokawa independently made the following conjecture:
For given Hecke eigenform f of weight 2k, with k odd, there should be a Siegel cusp form F ∈ Sk+1(Sp2(Z)), which is a Hecke eigenform, and its L-functions are given by
L(s, F,spin) =L(s, f)ζ(s−k)ζ(s−k+ 1), L(s, F,st) =ζ(s)L(s+k, f)L(s+k−1, f).
This conjecture was proved by Maass, Andrianov, Eichler-Zagier and Piatetski-Shapiro.
Around 1996, Duke and Imamoglu made the following conjecture, which is a generaliztion of Saito-Kurokawa conjecture:
For given Hecke eigenform f of weight 2k, with k congruent to n mod 2, there should be a Siegel cusp form F ∈Sk+n(Sp2n(Z)), which is a Hecke eigenform, and its standard L-function is given by
L(s, F,st) = ζ(s)
∏2n
i=1
L(s+k+n−i, f)
Ibukiyama also made a similar conjecture, which was formulated in terms of Koecher-Maass series.
§ Fourier coefficients of the Siegel Eisenstein series
For any positive definite half-integral symmetric matrix B of degree 2n, we put
DB = det(2B), dB =|Disc(Q(√
((−1)nDB)))|, fB =
√
DBd−B1 ∈N
1
Let χB be the primitive Dirichlet character corresponding to the quadratic extension Q(√
(−1)nDB)).
Fix a prime p. Let ψp be the unique additive character of Qp such that ψp(x) = exp(−2π√
−1εx) for x∈Z[p−1]. Note thatψp is of order 0. Recall that the Siegel series for B is defined by
bp(B, s) = ∑
R∈S2n(Qp)/S2n(Zp)
ψp(tr(BR))p−ordp(ν(R))s, Re(s)≫0.
Here,
S2n(Qp) :={R = tR|R ∈M2n(Qp)} S2n(Zp) :={R = tR|R ∈M2n(Zp)}
ν(R) := [RZ2np +Z2np :Z2np ].
Put
γp(B;X) = (1−X)(1−pnχB(p)X)−1
∏n
i=1
(1−p2iX2).
Then it is known that there exists a polynomialFp(B;X)∈Z[X] such that
bp(B, s) =γp(B;p−s)Fp(B;p−s).
Katsurada proved the functional equation
Fp(B;p−2n−1X−1) = (pn+(1/2)X)−2ordpfBFp(B;X).
In particular, degFp(B;X) = 2ordpfB.
Put ˜Fp(B;X) = X−ordpfBFp(B;p−n−(1/2)X). Then the functional equation reads
F˜p(B;X−1) = ˜Fp(B;X).
Assume thatk≫0 andk≡nmod 2. Consider the Siegel Eisenstein series of degree 2n:
E2n,k+n(Z) = ∑
(C,D)/∼
det(CZ+D)−k−n.
Here (C, D) extends over the equivalence classes of symmetric coprime pairs. We normalize the Eisenstein series as follows.
E2n,k+n(Z) = 2−nζ(1−2k−2n)
∏n
i=1
ζ(1 + 2i−2k−2n)E2n,k+n(Z).
If B is a positive definite half-integral symmetric matrix, then the calssical Fourier coefficient formula says that the B-th Fourier coef- ficient of E2n,k+n(Z) is equal to
L(χB,1−k)fkB−(1/2) ∏
p|DB
F˜p(B;pk−(1/2)).
§ Duke-Imamoglu conjecture Assume k ≡n mod 2. Let
f(τ) =
∑∞ N=1
a(N)qN ∈S2k(SL2(Z)), q=e2π√−1τ
be a normalized Hecke eigenform of weight 2k. TheL-function L(s, f) is defined by
L(s, f) =
∑∞ N=1
a(N)N−s
=∏
p
[1−a(p)X+p2k−1X2]−1
=∏
p
[(1−pk−(1/2)αpX)(1−pk−(1/2)α−p1X)]−1
Let Sk+(1/2)+ (Γ0(4)) be the Kohnen plus subspace of Sk+(1/2)(Γ0(4)) (See [6]). It is the space of modular forms of weight k + 12 whose N- th Fourier coefficients vanishes unless (−1)kN ≡ 0,1. The Shimura correspondence gives one-to-one correspondence between the set of Hecke eigenforms of S2k(SL2(Z)) and the set of Hecke eigenforms of the Kohnen subspace Sk+(1/2)+ (Γ0(4)), up to scalar. Let
h(τ) = ∑
N >0 εN≡0,1 (4)
c(N)qN ∈Sk+(1/2)+ (Γ0(4))
be a Hecke eigenform which corresponds to f(τ) by the Shimura cor- respondence.
For each positive definite half-integral symmetric matrix B of size 2n, we put
A(B) =c(dB)fkB−(1/2)∏
p
F˜p(B;αp).
Note that the right-hand side does not depend on the choice of αp. We define
F(Z) = ∑
B=tB>0
A(B)e(BZ), Z ∈h2n.
Then our first main theorem is as follows.
Theorem 1. Assume that k ≡ n mod 2. Then F(Z) is a non-zero Siegel cusp form of weight k +n and degree 2n. F(Z) is a Hecke eigenform and its standard L-function is given by
L(s, F,st) =ζ(s)
∏2n
i=1
L(s+k+n−i, f).
We shall callF(Z) the Duke-Imamoglu lift of f(τ) to degree 2n.
§ More construction Let
f(τ) =
∑∞ N=1
a(N)qN ∈S2k(SL2(Z)), q=e2π√−1τ be a normalized Hecke eigenform.
Now assume k ≡ r+n mod 2. By Theorem 1, we have a Hecke eigenform
F(Z)∈Sk+n+r(Sp2n+2r(Z)) whose standard L-function is equal to
ζ(s)
2n+2r∏
i=1
L(s+k+n+r−i, f).
Let g(Z) ∈ Sk+n+r(Spr(Z)) be a Hecke eigenform, whose standard L-function is L(s, g,st).
We define Ff,g(Z) =
∫
Spr(Z)\hr
F
((Z 0 0 Z′
))
g(Z′)(det ImZ′)k+n−1dZ′, Z ∈h2n+r. Then Ff,g ∈Sk+n+r(Sp2n+r(Z)).
Theorem 2. If Ff,g ̸≡0, then Ff,g is a Hecke eigenform, and L(s,Ff,g,st) =L(s, g,st)
∏2n
i=1
L(s+k+n−i, f).
§ Examples
Example 1: It is known that dimS8(Sp4(Z)) = 1. A cusp form in that space is known as Schottky form. Its zero locus is exactly the closure of the Jacobian locus. It is the Duke-Imamogulu lift of ∆(τ)∈ S12(SL2(Z)) to degree 4.
Example 2: By Borcherds, Freitag, and Weissauer [1], there exists a Siegel cusp form of degree 12 and weight 12
F(12)(Z)∈S12(Sp12(Z))
which is a linear combination of the theta series associated to rank 24 even unimodular lattices. Then F(12) is the Duke-Imamoglu lift of
∆(τ)∈S12(SL2(Z)) to degree 12.
Example 3: It is known that dimS12(Sp3(Z)) = 1. Choose a non-zero cusp form F(3)(Z)∈S12(Sp3(Z)). Then Theorem 2 implies
L(s, F(3),st) =L(s,∆,st)L(s+ 10, ϕ20)L(s+ 9, ϕ20).
Here ϕ20 ∈ S20(SL2(Z)) is normalized Hecke eigenform of weight 20, and L(s,∆,st) is the adjoint L-function of ∆(τ), which is of degree 3.
This equality was conjectured by Miyawaki [7] in 1992, based on his numerical calculation.
References
[1] R. E. Borcherds, E. Freitag, and R. Weissauer, A Siegel cusp form of degree 12 and weight 12, J. Reine Angew. Math.494(1998), 141–153.
[2] S. Breulmann and M. Kuss, On a conjecture of Duke-Imamoglu, Proc. Amer.
Math. Soc107(2000),
[3] M. Eichler and D. Zagier,The theory of Jacobi forms, Progress in Mathematics 55Birkh¨auser Boston, Inc., Boston, Mass. 1985.
[4] T. Ikeda, On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n, to appear in Ann. of Math.
[5] T. Ikeda, On some construction of Siegel cusp forms: Miyawaki’s conjecture, preprint.
[6] W. Kohnen, Modular forms of half-integral weight onΓ0(4), Math. Ann.248 (1980), 249–266.
[7] I. Miyawaki, Numerical examples of Siegel cusp forms of degree 3 and their zeta functions, Mem. Fac. Sci. Kyushu Univ.46(1992), 307–309.
[8] G. Nebe and B. Venkov,On Siegel modular forms of weight12, J. Reine Angew.
Math.531(2001), 49–60.
Graduate school of mathematics, Kyoto University, Kitashirakawa, Kyoto, 606-8502, Japan
E-mail address: [email protected]