Vector valued Siegel modular forms

with small levels

Hiroki Aoki （青木宏樹）

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

[Notations]

ℍ_{2} :=\{Z =^{t}Z =(\begin{matrix}τ & z \\ z & ω \end{matrix})∈M_{2}(ℂ) | Im Z >0\}

Sp_{4}(ℝ):=\{M =(\begin{matrix}A & B \\ C & D \end{matrix})∈M_{4}(ℝ) | ^{t}MJM =J\}

( J:=(\begin{matrix}0 & −E_{2} \\ E_{2} & 0 \end{matrix}) )

ℍ_{2} ∋Z ↦M⟨Z⟩:=(AZ + B)(CZ + D)^{−1} ∈ℍ_{2}

(F |_{k}M)(Z):=det(CZ + D)^{−k}F (M⟨Z⟩) Γ : finite index subgroup of Sp_{4}(ℤ)

[Definition]

F : ℍ_{2} → ℂ, k ∈ ℤ

F ∈ A_{k,0} ⟺ F (Z) = (F |_{k}M)(Z) for ∀M ∈ Γ

A_{ℤ,0} := ⨁_{k∈ℤ} A_{k,0}

[Setting]

Γ = Sp_{4}(ℤ), Γ_{0}(2), Γ_{0,ψ_{3}}(3), Γ_{0,ψ_{4}}(4)

Γ_{0}(N):=\{M =(\begin{matrix}A & B \\ C & D \end{matrix})∈Sp_{4}(ℤ) | C ≡ 0 (N)\}

Γ_{0,ψ_{3}}(3):=\{M =(\begin{matrix}A & B \\ C & D \end{matrix})∈Γ_{0}(3) | (\frac{−3}{det D})=1\}

Γ_{0,ψ_{4}}(4):=\{M =(\begin{matrix}A & B \\ C & D \end{matrix})∈Γ_{0}(4) | (\frac{−1}{det D})=1\}

CASE 1 : Γ = Sp_{4}(ℤ)

[Theorem] (Igusa)

∑_{k∈ℤ} (dim_{ℂ} A_{k,0}) x^{k}

= \frac{1 + x^{35}}{(1 − x^{4})(1 − x^{6})(1 − x^{10})(1 − x^{12})}

Generators: E_{4}, E_{6}, Δ_{10}, Δ_{12} and Δ_{35} [Theorem] (Ibukiyama)

Δ_{35} = det (\begin{matrix}4E_{4} & 6E_{6} & 10Δ_{10} & 12Δ_{12} \\ \frac{∂E_{4}}{∂τ} & \frac{∂E_{6}}{∂τ} & \frac{∂Δ_{10}}{∂τ} & \frac{∂Δ_{12}}{∂τ} \\ \frac{∂E_{4}}{∂z} & \frac{∂E_{6}}{∂z} & \frac{∂Δ_{10}}{∂z} & \frac{∂Δ_{12}}{∂z} \\ \frac{∂E_{4}}{∂ω} & \frac{∂E_{6}}{∂ω} & \frac{∂Δ_{10}}{∂ω} & \frac{∂Δ_{12}}{∂ω} \end{matrix})

(4 + 6 + 10 + 12 + 3 = 35)

CASE 2 : Γ = Γ_{0}(2) (Ibukiyama)

∑_{k∈ℤ} (dim_{ℂ} A_{k,0}) x^{k}

= \frac{1 + x^{19}}{(1 − x^{2})(1 − x^{4})(1 − x^{4})(1 − x^{6})}

CASE 3 : Γ = Γ_{0,ψ_{3}}(3) (Ibukiyama, A)

∑_{k∈ℤ} (dim_{ℂ} A_{k,0}) x^{k}

= \frac{1 + x^{14}}{(1 − x)(1 − x^{3})(1 − x^{3})(1 − x^{4})}

CASE 4 : Γ = Γ_{0,ψ_{4}}(4) (Hayashida, Ibukiyama)

∑_{k∈ℤ} (dim_{ℂ} A_{k,0}) x^{k}

= \frac{1 + x^{11}}{(1 − x)(1 − x^{2})(1 − x^{2})(1 − x^{3})}

[Definition]

F : ℍ_{2} → ℂ^{3}, k ∈ ℤ

F ∈ A_{k,2} ⟺ F (Z) = ρ(CZ + D)^{−1}(F |_{k}M)(Z) for ∀M ∈ Γ

ρ (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}a^{2} & ab & b^{2} \\ 2ac & ad + bc & 2bd \\ c^{2} & cd & d^{2} \end{matrix})

A_{ℤ,2} := ⨁_{k∈ℤ} A_{k,2}

A_{ℤ,2} is a graded module of A_{ℤ,0}.

A_{2ℤ,0} := ⨁_{k∈2ℤ} A_{k,0}

A_{ℤ,0} is a graded module of A_{2ℤ,0}. A_{ℤ,2} is a graded module of A_{2ℤ,0}.

CASE 1 : Γ = Sp_{4}(ℤ)

[Theorem] (Satoh)

∑_{k∈2ℤ} (dim_{ℂ} A_{k,2}) x^{k}

=\frac{x^{10} + x^{14} + x^{16}}{(1 − x^{4})(1 − x^{6})(1 − x^{10})(1 − x^{12})}
+\frac{x^{16} + x^{18}}{(1 − x^{6})(1 − x^{10})(1 − x^{12})} + \frac{x^{22}}{(1 − x^{10})(1 − x^{12})}

[E_{4}, E_{6}] := (\begin{matrix}4E_{4}\frac{∂E_{6}}{∂τ} − 6E_{6}\frac{∂E_{6}}{∂τ} \\ 4E_{4}\frac{∂E_{6}}{∂z} − 6E_{6}\frac{∂E_{6}}{∂z} \\ 4E_{4}\frac{∂E_{6}}{∂ω} − 6E_{6}\frac{∂E_{6}}{∂ω} \end{matrix})

10 = 4 + 6 16 = 6 + 10
14 = 4 + 10 18 = 6 + 12
16 = 4 + 12 22 = 10 + 12

[Theorem] (Ibukiyama)

∑_{k∈2ℤ+1} (dim_{ℂ} A_{k,2}) x^{k}

=\frac{x^{21} + x^{23} + x^{27} + x^{29} − x^{33}}{(1 − x^{4})(1 − x^{6})(1 − x^{10})(1 − x^{12})}

[E_{4}, E_{6}, Δ_{10}] :=

4E_{4}(\frac{∂E_{6}}{∂τ} \frac{∂Δ_{10}}{∂z} − \frac{∂Δ_{10}}{∂τ} \frac{∂E_{6}}{∂z})−6E_{6}(\frac{∂E_{4}}{∂τ} \frac{∂Δ_{10}}{∂z} + \frac{∂Δ_{10}}{∂τ} \frac{∂E_{4}}{∂z})+10Δ_{10}(\frac{∂E_{4}}{∂τ} \frac{∂E_{6}}{∂z} − \frac{∂E_{6}}{∂τ} \frac{∂E_{4}}{∂z}) 8E_{4}(\frac{∂E_{6}}{∂τ} \frac{∂Δ_{10}}{∂ω} − \frac{∂Δ_{10}}{∂τ} \frac{∂E_{6}}{∂ω})−12E_{6}(\frac{∂E_{4}}{∂τ} \frac{∂Δ_{10}}{∂ω} + \frac{∂Δ_{10}}{∂τ} \frac{∂E_{4}}{∂ω})+20Δ_{10}(\frac{∂E_{4}}{∂τ} \frac{∂E_{6}}{∂ω} − \frac{∂E_{6}}{∂τ} \frac{∂E_{4}}{∂ω}) 4E_{4}(\frac{∂E_{6}}{∂z} \frac{∂Δ_{10}}{∂ω} − \frac{∂Δ_{10}}{∂z} \frac{∂E_{6}}{∂ω})−6E_{6}(\frac{∂E_{4}}{∂z} \frac{∂Δ_{10}}{∂ω} + \frac{∂Δ_{10}}{∂z} \frac{∂E_{4}}{∂ω})+10Δ_{10}(\frac{∂E_{4}}{∂z} \frac{∂E_{6}}{∂ω} − \frac{∂E_{6}}{∂z} \frac{∂E_{4}}{∂ω})

21 = 4 + 6 + 10 + 1 23 = 4 + 6 + 12 + 1 27 = 4 + 10 + 12 + 1 29 = 6 + 10 + 12 + 1

4E_{4}[E_{6}, Δ_{10}, Δ_{12}] − 6E_{6}[E_{4}, Δ_{10}, Δ_{12}] + 10Δ_{10}[E_{4}, E_{6}, Δ_{12}] − 12Δ_{12}[E_{4}, E_{6}, Δ_{10}] = 0 CASE 2 : Γ = Γ_{0}(2)

6 = 2 + 4, 8 = 4 + 4′, 11 = 2 + 4 + 4′ + 1,
6 = 2 + 4′, 10 = 4 + 6, 13 = 2 + 4 + 6 + 1,
8 = 2 + 6, 10 = 4′ + 6, 13 = 2 + 4′ + 6 + 1,
15 = 4 + 4′ + 6 + 1

CASE 3 : Γ = Γ_{0,ψ_{3}}(3)

4 = 1 + 3, 6 = 3 + 3′, 8 = 1 + 3 + 3′ + 1,
4 = 1 + 3′, 7 = 3 + 4, 9 = 1 + 3 + 4 + 1,
5 = 1 + 4, 7 = 3′ + 4, 9 = 1 + 3′ + 4 + 1,
11 = 3 + 3′ + 4 + 1

CASE 4 : Γ = Γ_{0,ψ_{4}}(4)

3 = 1 + 2, 4 = 2 + 2′, 6 = 1 + 2 + 2′ + 1,
3 = 1 + 2′, 5 = 2 + 3, 7 = 1 + 2 + 3 + 1,
4 = 1 + 3, 5 = 2′ + 3, 7 = 1 + 2′ + 3 + 1,
8 = 2 + 2′ + 3 + 1

[Idea of the proof]

A_{k} : A_{k,0} or A_{k,2}

(D_{r}F) (τ, ω) := \frac{1}{(2π\sqrt{−1})^{r}} (\frac{∂^{r}F}{∂z^{r}} ) (\begin{matrix}τ & 0 \\ 0 & ω \end{matrix})

A_{k}[r] := {F ∈ A_{k}|D_{s}F = 0 if s < r}

0 ⟶ A_{k}[r + 1] ⟶ A_{k}[r] ⟶^{D_{r}} D_{r}(A_{k}[r]) ⟶ 0

dim_{ℂ} A_{k} = ∑_{r=0}^{∞} dim_{ℂ} D_{r}(A_{k}[r])

( k : fix ⇒ finite sum )

Image of D_{r} : modular form of wt. k + r w.r.t. τ modular form of wt. k + r w.r.t. ω vanish first some Fourier coefficients [Basic Case]

A_{k} = A_{k,0}, Γ = Sp_{4}(ℤ), k : even

r : odd ⇒ D_{r}(A_{k}[r]) = {0} r : even ⇒ Image of D_{r} :

SL_{2}(ℤ)-modular of wt. k + r w.r.t. τ SL_{2}(ℤ)-modular of wt. k + r w.r.t. ω invariant w.r.t τ ↔ ω vanish first some Fourier coefficients

D_{r}(A_{k}[r]) is a subspace of the space generated by

f_{4} := e_{4}(τ)e_{4}(ω) : weight (4, 4)
f_{6} := e_{6}(τ)e_{6}(ω) : weight (6, 6)
f_{12} := Δ(τ)Δ(ω) : weight (12, 12)

\frac{1}{(1 − x^{4})(1 − x^{6})(1 − x^{12})}

F (\begin{matrix}τ & z \\ z & ω \end{matrix}) = ∑_{4nm−l^{2}≥0} a(n, l, m)e(nτ + lz + mω),

( e(w) = exp(2π\sqrt{−1}w) )

(D_{s}F) (τ, ω) = ∑_{n,m} (∑_{l} a(n, l, m) l^{s}) e(nτ + mω)

a(n, l, m) =a(n, −l, m)
=a(m, l, n)
=a(n+tl+t^{2}m, l+2tm, m) (t ∈ ℤ)

⇒ a(n, l, m) is primitive if 0 ≤ l ≤ min(n, m)

F ∈ A_{k}[2r]

⇒ ∑_{l} a(n, l, m) l^{2s} = 0 for s = 0, 1, …, r − 1
⇒a(n, l, m) = 0 if min(n, m) < r

( by induction and Vandermonde )

D_{2r}(A_{k}[2r]) ⊂ (f_{12}^{r} ℂ[f_{4}, f_{6}, f_{12}])_{k+2r,k+2r}

∑_{k∈2ℤ} (dim_{ℂ} A_{k}) x^{k}

= ∑_{k∈2ℤ} ∑_{r=0}^{∞} dim_{ℂ} (D_{2r}(A_{k}[2r])) x^{k}

≦ ∑_{r=0}^{∞} \frac{x^{10r}}{(1 − x^{4})(1 − x^{6})(1 − x^{12})}

=\frac{1}{(1 − x^{4})(1 − x^{6})(1 − x^{10})(1 − x^{12})}

[Multi cusps]

A_{k} = A_{k,0}, Γ = Γ_{0}(2) k : even

dim_{ℂ} A_{k} = ∑_{r=0}^{∞} dim_{ℂ} D_{r}(A_{k}[r])

γ := \frac{1}{\sqrt{2}} (\begin{matrix}0 & −E_{2} \\ E_{2} & 0 \end{matrix})

F ∈ A_{k}[r] ⇔ F |_{k}γ ∈ A_{k}[r]

r : odd ⇒ D_{r}(A_{k}[r]) = {0} r : even ⇒ Image of D_{r} :

Γ_{0}(2)-modular of wt. k + r w.r.t. τ Γ_{0}(2)-modular of wt. k + r w.r.t. ω invariant w.r.t τ ↔ ω vanish first r Fourier coefficients at each cusps [With character]

A_{k} = A_{k,0}, Γ = Γ_{0,ψ_{3}}(3)

Γ_{0,ψ_{3}}(3) is a normal subgroup of Γ_{0}(3)

[Γ_{0}(3) : Γ_{0,ψ_{3}}(3)] = 2

Γ_{0}(3) = Γ_{0,ψ_{3}}(3) ∪ Γ_{0,ψ_{3}}(3)γ_{A}

( γ_{A} = (\begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}) , ψ_{3}(γ_{A}) = −1 )

A_{k}(Γ_{0,ψ_{3}}(3)) = A_{k}(Γ_{0}(3)) ⊕ A_{k}(Γ_{0}(3), ψ_{3})

[Vector valued case]

F (\begin{matrix}τ & z \\ z & ω \end{matrix})

=(−1)^{k} (\begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}) F (\begin{matrix}ω & z \\ z & τ \end{matrix})

= (\begin{matrix}\frac{1}{(cτ +d)^{k+2}} & \frac{−cz}{(cτ +d)^{k+2}} & \frac{(cz)^{2}}{(cτ +d)^{k+2}} \\ 0 & \frac{1}{(cτ +d)^{k+1}} & \frac{−2cz}{(cτ +d)^{k+1}} \\ 0 & 0 & \frac{1}{(cτ +d)^{k}} \end{matrix} ) F (\frac{z}{cτ +d} ω − \frac{cz^{2}}{cτ +d}^{\frac{aτ +b}{cτ +d}  \frac{cz}{cτ +d}})

= (\begin{matrix}1 & t & t^{2} \\ 0 & 1 & 2t \\ 0 & 0 & 1 \end{matrix}) F (\begin{matrix}τ & z + tτ \\ z + tτ & ω + 2tz + t^{2}τ \end{matrix} )

(t ∈ ℤ)

F (\begin{matrix}τ & z \\ z & ω \end{matrix}) = ∑_{4nm−l^{2}≥0} a(n, l, m)e(nτ + lz + mω),

a(n, l, m)

=(−1)^{k} (\begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix}) a(n, −l, m)

=(−1)^{k} (\begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}) a(m, l, n)

= (\begin{matrix}1 & t & t^{2} \\ 0 & 1 & 2t \\ 0 & 0 & 1 \end{matrix}) a(n + tl + t^{2}m, l + 2tm, m)

(t ∈ ℤ)

k + r ∈ 2ℤ ⇒ D_{r}(F) = (\begin{matrix}∗ \\ 0 \\ ∗ \end{matrix})

k + r ∈ 2ℤ + 1 ⇒ D_{r}(F) = (\begin{matrix}0 \\ ∗ \\ 0 \end{matrix})

(−1)^{k} (\begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix}) a(n, m, m)

=a(n, −m, m)

= (\begin{matrix}1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix}) a(n, m, m)