Unformatted text preview: 87 Here we use that I 6∈ Γ( N ) for N > 1. We know that the Riemann surface X (1) = H * / Γ(1) parametrizes isomorphism clases of elliptic curves. For any elliptic curve E we denote by N E the subgroup of Ntorsion points. If E = C / Λ we have N E = 1 N Λ / Λ Theorem 8.5. There is a natural bijective map between the set of points of X ( N ) = X ( N ) \ { cusps } and isomorhism classes of pairs ( E,φ ) , where E is an elliptic curve and φ : ( Z /N ) 2 → N E is an isomorphism of groups. Two pairs ( E,φ ) and ( E ,φ ) are called isomorphic if there exists an isomorphism f : E → E of elliptic curves such that f ◦ φ = φ . Proof. Let E = C / Λ. Then N E = 1 N Λ / Λ. An isomorphism φ : ( Z /N ) 2 → N E is defined by a choice of a basis in N E . A representative of a basis is an ordered pair of vectors ( a,b ) from Λ such that ( Na,Nb ) is a basis of Λ. Replacing E by an isomorphic curve, we may assume that Λ = Z + τ Z for some τ ∈ H and ( Na,Nb ) = (1 ,τ ). This)....
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This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.
 Fall '09
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