MODULAR FORMS-page54

MODULAR FORMS-page54 - 50 LECTURE 6 MODULAR FORMS of in the space of holomorphic functions on H dened by(g(z = |k g 1 Note that we switched here to

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50 LECTURE 6. MODULAR FORMS of Γ in the space of holomorphic functions on H defined by ρ ( g )( φ ( z )) = φ | k g - 1 . (6.6) Note that we switched here to g - 1 in order to get ρ ( gg 0 ) = ρ ( g ) ρ ( g 0 ) . It follows from the above that to check (6.1) for some subgroup Γ it is enough to verify it only for generators of Γ. Now we use the following: Lemma 6.1. The group G = P SL(2 , Z ) = SL(2 , Z ) / {±} is generated by the matrices S = 0 - 1 1 0 « , T = 1 1 0 1 « . These matrices satisfy the relations S 2 = 1 , ( ST ) 3 = 1 . Proof. We know that the modular figure D (more exactly its subset D 0 ) is a fundamen- tal domain for the action of G in the upper half-plane H by Moebius transformations. Take some interior point z 0 ∈ D and any g G . Let G 0 be the subgroup of G gener- ated by S and T . If we find an element g 0 G such that g 0 g · z 0 belongs to D , then g 0 g = 1 and hence g G 0 . Let us do it. First find g 0 G 0 such that Im ( g 0 · ( g · z 0 )) is maximal possible. We have, for any
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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.

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