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Unformatted text preview: 51 6.2 Suppose we have n + 1 linearly independent functions f ,...,f n satisfying (6.1) (with the same number k ). Then we can consider the map f : H → CP n , τ → ( f ( τ ) ,...,f n ( τ ) . (6.7) When we replace τ with ατ + β γτ + δ , the coordinates of the image will all multiply by the same number, and hence define the same point in the projective space. This shows that the map f factors through the map ¯ f : H / SL(2 , Z ) → CP n . Now recall that the points of H / SL(2 , Z ) are in a natural bijective correspondence with the isomorphism classes of elliptic curves. This allows us (under certain conditions) to view the set of elliptic curves as a subset of a projective space and study it by means of algebraic geometry. Other problems on elliptic curves lead us to consider the sets of elliptic curves with additional structure. These sets are parametrized by the quotient H / Γ where Γ is a subgroup of SL(2 , Z ) of finite index. To embed these quotients we need to consider functions satisfying property (6.1) but only restriced to matrices fromneed to consider functions satisfying property (6....
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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