Eta Products and Models for Modular Curves

Pullbacks by Level-Lowering Maps
Suppose that Ml ∣ ∣N. It is well-known that we have maps,
π l : X 0 (N) → X 0 (M), defined by
π l (E, C) = (E/C[l], C[Ml]/C[l]).
We can also define the π l pullback of a modular form f for Γ 0 (M).
Definition: Let f be a weight k modular form for Γ 0 (M), with Ml ∣ ∣N as
above. Then we define
(π ∗ l f )(E, C) = ι ∗ (f (E/C[l], C[Ml]/C[l])),
where ι : E → E/C[l] is the canonical isogeny and
ι ∗ : Ω ⊗k
E/C[l] → Ω⊗k E .