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By Knopp M.I. (ed.)

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The set of equivalence classes is denoted S0 (N ) = {enhanced elliptic curves for Γ0 (N )}/ ∼ . An element of S0 (N ) is denoted [E, C], the square brackets connoting equivalence class. An enhanced elliptic curve for Γ1 (N ) is a pair (E, Q) where E is a complex elliptic curve and Q is a point of E of order N . ) Two such pairs (E, Q) and (E , Q ) are equivalent if some ∼ isomorphism E −→ E takes Q to Q . The set of equivalence classes is denoted S1 (N ) = {enhanced elliptic curves for Γ1 (N )}/ ∼ .

Ad − bc − kN = 1 for some a, b, and k, and the matrix γ = ac db ∈ M2 (Z) reduces modulo N into SL2 (Z/N Z). Modifying the entries of γ modulo N doesn’t affect Q, so since SL2 (Z) surjects to SL2 (Z/N Z) we may take γ = a b ∈ SL (Z). Let τ = γ(τ ) and let m = cτ + d. 1 for the third equality), and m 1 + Λτ N = cτ + d + Λτ = Q. N This shows that [E, Q] = [C/Λτ , 1/N + Λτ ] where τ ∈ H. Suppose two points τ, τ ∈ H satisfy Γ1 (N )τ = Γ1 (N )τ . Thus τ = γ(τ ) where γ = ac db ∈ Γ1 (N ). Again let m = cτ + d.

4. Let Γ be a congruence subgroup of SL2 (Z) of level N , and let qN = e2πiτ /N for τ ∈ H. 3 and satisfies (3 ) In the Fourier expansion f (τ ) = satisfy the condition |an | ≤ Cnr ∞ n=0 n an qN , the coefficients for n > 0 for some positive constants C and r. 3, and so f ∈ Mk (Γ ). For instance, the proposition easily shows that θ(τ, 4) ∈ M2 (Γ0 (4)). 9. 3 reduces to previous ones when Γ = SL2 (Z). As before, the modular forms and the cusp forms are vector spaces and subspaces, and the sums M(Γ ) = Mk (Γ ) and S(Γ ) = Sk (Γ ) k∈Z k∈Z form a graded ring and a graded ideal.

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