MODULAR FORMS-page91

MODULAR FORMS-page91 - 87 Here we use that- I 6 ( N ) for N...

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Unformatted text preview: 87 Here we use that- I 6 ( N ) for N > 1. We know that the Riemann surface X (1) = H * / (1) parametrizes isomorphism clases of elliptic curves. For any elliptic curve E we denote by N E the subgroup of N-torsion points. If E = C / we have N E = 1 N / 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, ) , where E is an elliptic curve and : ( Z /N ) 2 N E is an isomorphism of groups. Two pairs ( E, ) and ( E , ) are called isomorphic if there exists an isomorphism f : E E of elliptic curves such that f = . Proof. Let E = C / . Then N E = 1 N / . An isomorphism : ( 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 such that ( Na,Nb ) is a basis of . Replacing E by an isomorphic curve, we may assume that = Z + Z for some H and ( Na,Nb ) = (1 , ). This)....
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