Some Graded Rings of Hermitian Modular Forms

Some Graded Rings of Hermitian Modular Forms

by Aloys Krieg

In the early 1960’s Igusa [8] determined the graded ring of Siegel modular forms of degree 2. In this paper, which is based on the recent thesis of Dern [3], we will describe the graded rings of Hermitian modular forms of degree 2 overQ(√

−1),Q(√

−2) andQ(√

−3). Therefore we will determine all abelian characters of the Hermitian modular groups. We will describe all Maa lifts and construct appropriate Borcherds products. Finally we proceed along the lines of Igusa’s reduction process.

LetK be an imaginary quadratic number field with discriminantdK and ring of integers OK. Then the Hermitian modular group of degree 2 over K is

Γ₂(K) :=

{

M ∈^O4K^×4; M J M^tr =J }

, J =

(0 −I I 0

)

, I = (1 0

0 1 )

,

wheretrstands for the transpose. One has Γ₂(K)⊂SL₄(K) ifd_K ̸=−3,−4.

Theorem 1 ([2]). Γ₂(K)∩SL₄(K) possesses a non-trivial abelian character if and only if d_K is even. In this case it is uniquely determined and denoted by χ_K.

χ_K is an extension of the non-trivial abelian character on the Siegel modular group of degree 2.

The Hermitian half-space of degree 2 is given by H₂(C) :=

{

Z ∈C^2×2; 1 2i

(

Z −Z^tr )

positive definite }

.

It contains the Siegel half-space H₂(R) := {Z ∈H₂(C); Z =Z^tr} as a sub- manifold. If Γ ⊂ Γ₂(K) is a subgroup of finite index and ν is an abelian character of Γ then the space [Γ, k, ν], k ∈Z, of Hermitian modular forms of weightkand characterνin the sense of Hel Braun consists of all holomorphic functions f :H₂(C)→Csatisfying

f(

(AZ+B)(CZ+D)⁻¹)

=ν(M) det(CZ+D)^kf(Z) forM = (^{A B}_{C D})∈Γ.

The transpose mapping is an exceptional automorphism of H2(C). Hence the subspace [Γ, k, ν]_sym resp. [Γ, k, ν]_skew consists of all such f satisfying

f( Z^tr)

=f(Z) resp. f( Z^tr)

=−f(Z).

Note that any skew-symmetric f vanishes on H₂(R).

Examples of Hermitian modular forms are given by the Eisenstein series E_k(Z) = ∑

M:(^{∗ ∗}_{0 ∗})\Γ2(K)

(detM)^k/2det(CZ+D)^−k∈[

Γ₂(K), k,det^−k/2 ]

sym

for any evenk >4. MoreoverE₄ ∈[

Γ₂(K),4,det⁻²]

symcan be defined either by the Hecke trick or as a Maa lift or as a theta series.

Considering K = Q(√

−3) the group of abelian characters of Γ₂(K) is of order 3 generated by the determinant. If Γ = Γ₂(K)∩SL₄(K) any Hermitian modular form f possesses a Fourier and Fourier-Jacobi expansion

f(Z) = ∑

T=

(n λ λ m

)

≥0 n,m∈N0,λ∈√1−3OK

α_f(T)e^{2πi trace(T Z)}

=

∑∞ m=0

φ_m(τ, z, w)e^2πimω, Z =

(τ z w ω

) ,

whereφ_m is a Jacobi form of weightk and index m (cf. [5], [11]). Note that f is (skew-) symmetric if and only if

φ_m(τ, w, z) = φ_m(τ, z, w) (resp.φ_m(τ, w, z) =−φ_m(τ, z, w)) for all m∈N0. Theorem 2([3], [5], [9], [11]). Let K =Q(√

−3).

a) The Maa space M^sym_k attached to the symmetric Jacobi forms of weight k and index 1 is contained in

[

Γ₂(K), k,det^−k/2 ]

, k even, and isomorphic to the Maa Spezialschar. One has

E_k∈ M^sym_k , k ≥4even.

b) The Maa space M^skewk attached to the skew-symmetric Jacobi forms of weight k and index 1 is contained in [

Γ₂(K), k,det^k]

, k odd, and isomorphic to the space of elliptic modular forms

[SL₂(Z), k−9,1].

There exists a non-trivial ϕ₉ ∈ M^skew9 .

Now we consider modular forms with respect to the split orthogonal group O(2,4) or more precisely the connected component O⁺(2,4) of the identity in the realization on the quadratic space (V, q), where

V =R²×R²×C, q(l₁, l₂, l₃, l₄, λ) = l₁l₂+l₃l₄− |λ|².

The half-space attached to O⁺(2,4) can be mapped bijectively onto H₂(C).

Considering the particular lattice

L=Z²×Z²×OK

we obtain the modular group

{σ∈O⁺(2,4);σ(L) = L}/{±1},

which is isomorphic to the group generated by the modular transformations of Γ₂(K) and Z 7→ Z^tr. Hence we can identify Borcherds products with Hermitian modular forms. In particular ϕ9 turns out to be a Borcherds product with zeros of 1. order exactly on ∪

M∈Γ2(K)M⟨H₂(R)⟩. Moreover a Borcherds product ϕ₄₅ ∈[Γ₂(K),45,1]_sym exists, which vanishes of 1. order exactly on ∪

M∈Γ2(K)M⟨H⟩,H ={Z ∈H₂(C);z =−w}. The result is Theorem 3([3]). If K =Q(√

−3)the graded ring ⊕k[Γ₂(K)∩SL₄(K), k,1]

of Hermitian modular forms is generated by E₄, E₆, ϕ₉, E₁₀, E₁₂, ϕ₄₅.

Here E4, E6, ϕ9, E10, E12 are algebraically independent and the only relation is given by

ϕ²₄₅∈C[E₄, E₆, ϕ₉, E₁₀, E₁₂].

From this result it is easy to determine related rings, e.g.

⊕k[Γ₂(K), k,1]_sym =C[E₆, E₄³, E₁₂, E₄²E₁₀, ϕ²₉, E₄E₁₀² , E₁₀³ , ϕ₄₅],

and the field of Hermitian modular functions.

ConsideringK =Q(√

−1) there exists the character χ=χ_K apart from det such that the commutator subgroupCΓ₂(K) has the index 4. In this case we can analogously construct Borcherds products

ϕ₄ ∈[Γ₂(K),4, χdet]_skew, ϕ₁₀∈[Γ₂(K),10,1]_sym, ϕ₃₀∈[Γ₂(K),30, χdet]_sym, whereϕ₄ and ϕ₁₀ are also Maa lifts.

Theorem 4 ([1], [3], [4], [6], [7]). If K = Q(√

−1) then the graded ring

⊕k[CΓ₂(K), k,1] of Hermitian modular forms is generated by ϕ₄, E₄, E₆, ϕ₁₀, E₁₀, E₁₂, ϕ₃₀.

Here ϕ₄, E₄, E₆, E₁₀, E₁₂ are algebraically independent and the only relations are given by

ϕ²₁₀, ϕ²₃₀∈C[ϕ₄, E₄, E₆, E₁₀, E₁₂].

One has ϕ10

H2(R) = ∆²₅ and ϕ30

H2(R) = ∆30, where ∆5 and ∆30 are given by theta series in Igusa’s theorem [8]. The first skew-symmetric Hermitian modular form with trivial character H₃₄ constructed in a different way by Aoki [1] is equal toϕ₄·ϕ₃₀ up to a constant.

Considering K = Q(√

−2) things are more complicated since there do not exist Borcherds products with such simple divisor sets. In this case one has to lift generators of the graded rings of paramodular forms of level (^{1 0}_{0 2}) and (^{1 0}_{0 3}). There exist Borcherds products ϕ_k and modular forms f_k of weight k, whose definitions involve skew-symmetric Hermitian modular forms such that we obtain

Theorem 5 ([3]). If K = Q(√

−2) the graded ring ⊕k[CΓ₂(K), k,1]_sym of symmetric Hermitian modular forms is generated by

ϕ3, E4, f5, E6, ϕ8, f10, E12, ϕ24, f26, f28, f30.

Bibliography

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Preprint, 16 pages (2000).

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Aloys Krieg

Lehrstuhl A f¨ur Mathematik RWTH Aachen

D-52056 Aachen, Germany krieg@mathA.rwth-aachen.de