f ∈ S_{2}(Γ_{0}(N))new J_{0}(N) X_{0}(N) pN α, β : X_{0}(pN)X_{0}(N) X_{0}(N) (E, C)

E C E()

N ? (E, C)

C pN

pN

C ≅ C′⊕ D C′ N D p

C′ D α : (E, C) ↦ (E, C′)

β : (E, C) ↦ (E/D, (C+D)/D)

X_{0}(N) X_{0}(pN) Γ_{0}(N)\h Γ_{0}(pN)\h Γ_{0}(pN) ⊂ Γ_{0}(N) α′ : Γ_{0}(N)\hΓ_{0}(pN)\h.

α α′ β

Γ_{0}(pN) ↪ Γ_{0}(N)

X ↦ [\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}]·X·[\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}]^{−1} .

Γ_{0}(pN)\h ≃ [\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}] Γ_{0}(pN) [\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}]^{−1} \hΓ_{0}(N).

0(pN)[dl]_{α}[dr]^{β} X_{0}(N)X_{0}(N) α^{∗} : Div(X_{0}(N))Div(X_{0}(pN)).

(E, C) α (E, C⊕

D)

D E p

α^{∗} : (E, C) ↦ ∑_{|D|}(E, C⊕D),

E p

β_{∗} : X_{0}(pN)X_{0}(N) (E, C′) X_{0}(pN)

C′ C⊕ D C D E N p

β_{∗} : (E, C⊕D) ↦ (E/D, (C+D)/D). φ := β_{∗}∘

α^{∗} φ φ :

Div(X_{0}(N))Div(X_{0}(N)) φ α

Div^{0}(X_{0}(N))Div^{0}(X_{0}(N)) T_{p} J_{0}(N) T_{p} := φ|_{Div^{0}(X_{0}(N))} S_{2}(Γ_{0}(N)) ≅ H^{0}(X_{0}(N), Ω^{1}) f(z) ↦ f(z)dz f(z) Γ_{0}(N) f(z)dz

Γ_{0}(N) α β

H^{0}(X_{0}(N), Ω^{1})α^{∗}H^{0}(X_{0}(pN), Ω^{1})β_{∗}H^{0}(X_{0}(N), Ω^{1}).

T_{p} S_{2}(Γ_{0}(N)) pN X_{0}(N) J_{0}(N)() = H^{0}(X_{0}(N)(),)^{∗}/H_{1}(X_{0}(N)(),)

A_{f} J_{0}(N) J_{0}(N) H^{0}(X_{0}(N), Ω_{1})