On mixed weight Hilbert modular forms

Hiroki Aoki （青木宏樹）

Tokyo University of Science （東京理科大学・理工学部）

[Notations] K := ℚ(\sqrt{5}) quadratic field

O = ℤ [\frac{1+\sqrt{5}}{2}] integer ring

ε := \frac{1+\sqrt{5}}{2} fundamental unit

For x = u + v\sqrt{5} ∈ K, x′ := u − v\sqrt{5} ∈ K tr(x) := x + x′ = 2u ∈ ℚ

\begin{matrix}O^{∗} := & \{ν ∈ K| tr(μν) ∈ ℤ (∀μ ∈ O)\} \\ = & \{\frac{1}{2}u + \frac{\sqrt{5}}{10}v ∈ K | {u, v ∈ ℤ \atop u + v ∈ 2ℤ}\} \end{matrix}

ℍ := {τ ∈ ℂ | Im(τ) > 0} [Definition] For k_{1}, k_{2} ∈ ℤ, we say that a holomorphic function F on ℍ^{2} is a Hilbert modular form of weight (k_{1}, k_{2}) if

F (τ_{1}, τ_{2}) = (cτ_{1} + d)^{−k_{1}} (c′τ_{2} + d′)^{−k_{2}}

×F (\frac{aτ_{1} + b}{cτ_{1} + d}, \frac{a′τ_{2} + b′}{c′τ_{2} + d′})

for any ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(O).

[Notation]

k := \frac{k_{1} + k_{2}}{2} ∈ ℤ l := \frac{k_{1} − k_{2}}{2} ∈ ℕ_{0}

A_{k,l} : The space of all Hilbert modular forms of weight (k_{1}, k_{2})

[Question 1] Determine the structure of

A_{∗,∗} := ⨁_{k,l} A_{k,l}

as a bigraded ring

[Question 2] Determine the structure of

A_{∗,l} := ⨁_{k} A_{k,l}

as an A_{∗,0}-module.

[Fact] Gundlach determined the structure of A_{∗,0}. A_{∗,0} := R ⊕ G_{15}R, where R = ℂ[G_{2}, G_{5}, G_{6}].

[Koecher principle] F ∈ A_{k,l} has a Fourier expansion

F (τ_{1}, τ_{2}) = ∑_{\begin{matrix}ν∈O^{∗} \\ ν≥0, ν′≥0 \end{matrix}} c(ν) e (ντ_{1} + ν′τ_{2}),

where e(z) = exp(2π\sqrt{−1}z).

c(u, v) := c (\frac{1}{2}u + \frac{\sqrt{5}}{10}v)

F (τ_{1}, τ_{2}) = ∑_{u=0}^{∞} ∑_{|v|≤\sqrt{5}u} c(u, v)

×e ((\frac{1}{2}u + \frac{\sqrt{5}}{10}v) τ_{1} + (\frac{1}{2}u − \frac{\sqrt{5}}{10}v) τ_{2})

[Diagonal Differential Operator]

D := \frac{\sqrt{5}}{2π\sqrt{−1}} (\frac{∂}{∂τ_{1}} − \frac{∂}{∂τ_{2}}) (n ≥ 0)

(D_{n}F) (τ) := (D^{n}F) (τ, τ)

(D_{n}F) (τ)

= ∑_{u=0}^{∞} ( ∑_{|v|≤\sqrt{5}u} v^{n}c(u, v)) e(uτ)

A_{k,l}(n) := {F ∈ A_{k,l}|D_{r}(F) = 0 (r < n)} [Elliptic modular forms] M_{k} : The space of all elliptic modular forms of weight k w.r.t. SL_{2}(ℤ)

M_{k}(n)

:={f = ∑_{n=0}^{∞} a_{n}e(nτ)∈M_{k} |a_{r} =0 (r < n)}

⨁_{k∈ℤ} M_{k} = ℂ[e_{4}, e_{6}]

M_{k}(n) = Δ^{n}M_{k−12n}

e_{4} ∈ M_{4} e_{6} ∈ M_{6}

Δ = \frac{1}{1728} (e_{4}^{3} − e_{6}^{2}) ∈ M_{12}(1)

Lemma.

F ∈ A_{k,l}(n) ⇒ D_{n}F ∈ M_{2k+2n}.

Proposition. For a non-negative integer n, there exists an exact sequence

0 ⟶ A_{k,l}(n + 1) ⟶ A_{k,l}(n) ⟶^{D_{n}} M_{2k+2n}.

[Fourier coefficients]

F (τ_{1}, τ_{2}) = ∑_{\begin{matrix}ν∈O^{∗} \\ ν≥0, ν′≥0 \end{matrix}} c(ν) e (ντ_{1} + ν′τ_{2})

F ∈ A_{k,l} ⇒ c(ν) = ε^{−k}ε′^{−l}c(ε^{2}ν)

( Hence l ≠ 0 ⇒ c(0) = 0. )

c(u, v) := c (\frac{1}{2}u + \frac{\sqrt{5}}{10}v)

l ≠ 0, |v| ≥ u ⇒ ∃u_{0}, ∃v_{0} s.t.

u_{0} < u, c(u_{0}, v_{0}) = C · c(u, v)

( C ≠ 0 )

(D_{r}F) (τ)

= ∑_{u=0}^{∞} ( ∑_{|v|≤\sqrt{5}u} v^{r}c(u, v)) e(uτ)

[Fourier coefficients exist here !!]

F ∈ A_{k,l} ν = \frac{1}{2}u + \frac{\sqrt{5}}{10}v

[figure: u; O; 1; 2; 3; 4; 5; v; −7; −6; −5; −4; −3; −2; −1; 1; 2; 3; 4; 5; 6; 7; □; ∘; ∘; ∘; •; ∘; ∘; •; •; ∘; ∘; •; •; •; ∘; ∘; •; •; •; •; ∘; ⋆; ⋆; ⋆; ⋆; ⋆; ⋆; ⋆; ⋆; ⋆; ⋆; ⋆; ⋆; v = u; v = −u; v = \sqrt{5}u; v = −\sqrt{5}u; c(ν) = ε^{−k}ε′^{−l}c(ε^{2}ν)]

[Estimation] Lemma.

D_{n} (A_{k,l}(n)) ⊂ M_{2k+2n}(n + 1) (l ≠ 0)

Proposition.

dim A_{k,l} ≤ ∑_{n=0}^{∞} dim M_{2k+2n}(n + 1) (l ≠ 0)

In this proposition, the Poincare series of the right-hand-side is

\frac{x^{6}}{(1 − x^{2})(1 − x^{3})(1 − x^{5})}

=\frac{x^{6} + x^{9}}{(1 − x^{2})(1 − x^{5})(1 − x^{6})}

[Rankin-Cohen operators] M. H. Lee gave the way to construct mixed weight Hilbert modular forms by using generalized Rankin-Cohen operators.

[F_{1}, F_{2}]_{1} :=\frac{1}{2π\sqrt{−1}} (k_{1}F_{1} \frac{∂F_{2}}{∂τ_{1}} − k_{2}F_{2} \frac{∂F_{1}}{∂τ_{1}} ) ∈ A_{k_{1}+k_{2}+1,1}

and

[F_{1}, F_{2}]_{2} :=\frac{−1}{4π^{2}} (k_{1}(k_{1} + 1)F_{1} \frac{∂^{2}F_{2}}{∂τ_{1}^{2}} − 2(k_{1} + 1)(k_{2} + 1)\frac{∂F_{1}}{∂τ_{1}} \frac{∂F_{2}}{∂τ_{1}}

+k_{2}(k_{2} + 1)F_{2} \frac{∂^{2}F_{1}}{∂τ_{1}^{2}} ) ∈ A_{k_{1}+k_{2}+2,2}.

Jacobi identity

k_{1}F_{1} [F_{2}, F_{3}]_{1} + k_{2}F_{2} [F_{3}, F_{1}]_{1}

+k_{3}F_{3} [F_{1}, F_{2}]_{1} = 0.

[Construction]

G_{2} ∈ A_{2,0} : Symmetric
G_{5} ∈ A_{5,0} : Skew-symmetric
G_{6} ∈ A_{6,0} : Symmetric
G_{15} ∈ A_{15,0} : Symmetric

Theorem. (Gundlach)

{A_{∗,0} = R ⊕ RG_{15}. \atop R := ℂ[G_{2}, G_{5}, G_{6}].}

[Case l = 2]

\begin{matrix}G_{6,2} := [G_{2}, G_{2}]_{2} ∈ A_{6,2} \\ G_{9,2} := [G_{2}, G_{5}]_{2} ∈ A_{9,2} \end{matrix}

Theorem.

A_{∗,2} = RG_{6,2} ⊕ RG_{9,2}.

[Case l = 1] G_{8,1} := [G_{2}, G_{5}]_{1} ∈ A_{8,1} G_{9,1} := [G_{2}, G_{6}]_{1} ∈ A_{9,1} G_{12,1} := [G_{5}, G_{6}]_{1} ∈ A_{12,1}

Theorem.

A_{∗,1} = ℂ[G_{2}, G_{5}]G_{8,1} ⊕ RG_{9,1} ⊕ RG_{12,1}.