MODULAR FORMS-page59 - 55 6.4 We know that any elliptic...

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55 6.4 We know that any elliptic curve is isomorphic to a Hesse cubic curve. Let us give another cubic equation for an elliptic curve, called a Weierstrass equation. Its coefficients will give us new examples of modular forms. Recall that dim Th( k, Λ τ ) ab = k . Let use <, > to denote the linear span. We have Th(1 , Λ τ ) 1 2 1 2 = < ϑ 1 2 1 2 ( z ; τ ) > = < T > ; Th(2 , Λ τ ) = < T 2 , X >, Th(3 , Λ τ ) 1 2 1 2 = < T 3 , TX , Y >, for some functions X Th(2 , Λ τ ) , Y Th(3 , Λ τ ) 1 2 1 2 . Now the following seven func- tions T 6 , T 4 X , T 2 X 2 , X 3 , T 3 Y , TX Y , Y 2 all belong to the space Th(6 , Λ τ ) . They must be linearly dependent and we have aT 6 + bT 4 X 2 + cT 2 X 2 + dX 3 + eT 3 Y + fTX Y + gY 2 = 0 . (6.12) Assume g = 0 , d = 0. It is easy to find X = αX + βT 2 , Y = γY + δXT + ωT 3 which reduces this expression to the form Y 2 T - X 3 - AXT 4 - BT 6 = 0 , (6.13) for some scalars A, B . Let ( z ) = X/T 2 , 1 ( z ) = Y/T 3 . Dividing (6.13) by T 6 we obtain a relation 1 ( z ) 2 = ( z ) 3 + A℘ ( z ) + B. (6.14) Since both X and T 2 belong to the same space Th(2 , τ ) the functions ( z ) , ℘ 1 ( z ) have periods λ Z + τ Z and meromorphic on C . As we shall see a little later, 1 ( z ) = d℘ dz . Consider the map E τ = C / Λ P 2 , z
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