MODULAR FORMS-page99

MODULAR FORMS-page99 - 95 9.2 In view of this theorem the...

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Unformatted text preview: 95 9.2 In view of this theorem the cross-ratio R can be thought as a function R : H / Γ(2) → C . The next theorem shows that this function extends to a meromorphic function on X (2) = H * / Γ(2): Theorem 9.3. The cross-ratio function R extends to a meromorphic function λ on X (2) which generates the field M ( X (2)) . It can be explicitly given by the formula λ ( τ ) = ϑ 1 2 (0; τ ) 4 /ϑ 00 (0; τ ) 4 . Proof. It follows from the previous discussion that, as a function on H , the cross-ratio is given by R = R ( ℘ ( τ 2 + 1 2 ) ,℘ ( 1 2 ) , ∞ ,℘ ( τ 2 )) = ℘ ( τ 2 + 1 2 )- ℘ ( 1 2 ) ℘ ( τ 2 )- ℘ ( 1 2 + τ 2 ) . (9.5) We have dim M k (Γ(2)) = 1- 2 k + kμ 2 / 2 = k + 1 . (9.6) In particular dim M 1 (Γ(2)) = 2 . We have seen in Lecture 6 that ϑ 4 00 ,ϑ 4 1 2 ,ϑ 4 1 2 and ℘ ( τ 2 ) ,℘ ( τ 2 ) ,℘ ( 1 2 ) are examples of modular forms of weight 1 with respect to the group Γ(2). There must be some linear relation between these functions. The explicit relation between the first set is knownrelation between these functions....
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