Dimensions of cusp forms on U(2,1) for the Picard modular groups of imaginary quadratic fields

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Dimensions of cusp forms on U(2,1) for the Picard modular groups of imaginary quadratic fields

Ki-ichiro HASHIMOTO and Harutaka KOSEKI

The purpose of this note is to give an explicit formula for the dimension of the spaces of holomorphic cusp forms on the two dimensional bounded domain D which is holomorphically equivalent to the unit ball in C², with respect to the the Picard modular groups of imaginary quadratic fields. The result for certain discrete subgroup Γ of SU(1,2) has been calculated by several authors ([Ka], [Koj], [Sa]). However the case for full modular group has been known only for K =Q(√

−1) ([Co]).

Let K be an imaginary quadratic field and ρ be the nontrivial automorphism of K/Q. On V =K³ we fix two non-degenerate ρ-hermitian forms H and the unitary groups G= U(V, H) by

H(x, y) :=^ty^ρHx=x₁y₃^ρ−x₂y^ρ₂+x₃y₁^ρ with H=





0 0 1

0 −1 0

1 0 0



,

G_Q={g∈GL₃(K);^tg^ρHg=H},

LetO=OK be the ring of integers of K. The adelized groupGAacts on the set ofO-lattices in the natural way, and one can speak ofG-genera ofO-lattices in V. AnO-lattice Lis said to be unimodularif the dual lattice of L with respect toH coincides with L. Unimodular O-lattices in (V, H) form a single G-genus (cf. [Ja]).

The real Lie group G_∞ = G_R is connected, and acts on a Hermitian symmetric domain D; namely

D={z=^t(z, w)∈C²; 2Re(z)− |w|²>0}, g

( z 1

)

=j(g, z) (

g·z 1

)

(g∈G_∞, z ∈ D).

The second equality defines the elementg·zofDand the (scalar valued) automorphic factor j(g, z). We fix an origin z₀ of Dby

z0 = ( 1/2

0 )

(1) and denote byU_∞ the stabilizer ofz0 ∈ D inG_∞. ThenU_∞is a maximal compact subgroup of G. For a unimodular O-lattice Lin (V, H) we put

U_A(L) :=U_∞×U_A_f(L) with U_A_f := ^∏

p̸=∞

U_p(L_p).

Letk be a positive integer. AC-valued function f on the adelized groupGA is called a (scalar valued) holomorphic cusp form of weight k with respect to U_A(L), if it satisfies the following conditions:

(i) f is leftG_Q-, and right UA(L)-invariant, 1

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(ii) j(g_∞, z₀)^kf(g_∞g_f) depends only on z =g_∞·z₀ ∈ D and g_f; and it is holomorphic in z for any fixed gf ∈GAf,

(iii) f is bounded onG_A.

LetS_k(U_A(L)) denote theC-vector space of all such functions.

Theorem 0.1 Suppose thatKis an imaginary quadratic field with discriminantd(K)̸=−3,−4,−7,−8.

Then for even positive integer k > 4, the dimension of the space S_k(U_A) of holomorphic cusp- forms of weight k with respect toUA is given by

dimS_k(U_A) = 2T₁+ 2T_2,he+ 2T_2,es+ 2T₃⁽⁺⁾+ 2T₃⁽⁻⁾

+2T₄₁⁽⁺⁾+ 2T₄₁⁽⁻¹⁾2T₄₂⁽⁺⁾+ 2T₄₂⁽⁻¹⁾+ 2Tcu+ 2Tqu, with

T₁ = 1

2^t+53²h(K)B_3,χ(k−1)(k−2) T_2,he = 1

2^t+63h(K)²(2k−3)×







4|d(K)|+ 4 d(K)≡1 (mod 8) 4|d(K)| −2 d(K)≡5 (mod 8) 4|d(K)|+ 1 d(K)≡0 (mod 4).

T2,es = 1

2^t+63h(K)²×







4|d(K)| −4 d(K)≡1 (mod 8) 4|d(K)|+ 2 d(K)≡5 (mod 8) 4|d(K)| −1 d(K)≡0 (mod 4).

T₃⁽⁺⁾ = 1

2^t+4h(K)²×







4 d(K)≡1 (mod 8) 2 d(K)≡5 (mod 8) 3 d(K)≡0 (mod 4).

T₃⁽⁻⁾ = 1

2^t+4h(K)²×







0 d(K)≡1 (mod 8) 6 d(K)≡5 (mod 8) 3 d(K)≡0 (mod 4)







× {

1 k≡0,3 (mod 4)

−1 k≡1,2 (mod 4).

}

T₄₁⁽⁺⁾ = 1

2^t+23²h(K)²×







T₄₁⁽⁻⁾ = 1

2^t+23²h(K)²×













× {

2 k≡0 (mod 3)

−1 k≡1,2 (mod 3).

}

T₄₂⁽⁺⁾ = 1

2^t+23h(K)²×







T₄₂⁽⁻⁾ = 1

2^t+23h(K)²×











 2

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×







1 k≡1 (mod 3)

−1 k≡2 (mod 3) 0 k≡0 (mod 3).





 Tcu = − 1

2^t+33h(K)²×

{ 1 d(K)≡1,3 (mod 4) 4 d(K)≡0 (mod 4).

Tqu = − 1

2^t+1h(K)².

where h(K), B_3,χ denote the class number of K, the 3-rd generalized Bernoulli number with respect to the Dirichlet character attached to K, respectively.

Remark The dimension of the spaceSk(SUA) of cusp forms w.r.t. the special unitary group is obtained simply by dividing dimS_k(U_A) by h(K). Similar results for the cases d(K) =

−3,−4,−7,−8, as well as the dimension for vector valued cusp forms, have been obtained.

The details will appear elsewhere.

References

[Co] L.Cohn : The dimension of spaces of automorphic forms on a certain two dimensional complex domain, Memoirs Amer.Math.Soc., 158 (1975).

[HK1] K.Hashimoto and H.Koseki : Class numbers of definite unimodular hermitian forms over the ring of imaginary quadratic fields, Tˆohoku Math.J. 41 (1989), 1-30.

[HK2] K.Hashimoto and H.Koseki : Class numbers of positive definite binary and ternary unimodular hermitian forms, Tˆohoku Math.J. 41 (1989), 171-216.

[Ja] R.Jacobowitz : Hermitian forms over local fields, Amer.J.Math. 84 (1962), 441-465.

[Ka] S.Kato : A dimension formula for a certain space of automorphic forms of SU(p,1), Math.Ann. 266 (1984), 457-477.

[Kos] H.Koseki : On a comparison of trace formula for GU(1,2) and GU(3), J.Fac.Sci.Univ. of Tokyo 33 (1986), 467-521.

[Koj] H.Kojima :Dimension of spaces of vector valued automorphic forms on the unitary group SU(2,1), Tˆohoku Math.J. 36 (1984), 563-579.

[Sa] I.Satake : On numerical invariants of arithmetic varieties of Q-rank one, Automorphic forms of several variables, Taniguchi Symposium 1983, Katata, (1984),353-369.

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