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MODULAR FORMS-page56

MODULAR FORMS-page56 - 52 LECTURE 6 MODULAR FORMS...

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Unformatted text preview: 52 LECTURE 6. MODULAR FORMS represented by a rational number x or ∞ the stabilizer group Γ x is conjugate to a subgroup of SL(2 , Z ) ∞ . In fact, if g · x = ∞ for some g ∈ SL(2 , Z ), then g · Γ x · g- 1 · ∞ = ∞ . Since ατ + β γτ + δ · ∞ = ∞ ⇔ γ = 0 , we have g · Γ x · g- 1 ⊂ {± „ 1 β 1 « ,β ∈ Z } Let h be the smallest positive β occured in this way. Then it is immediately seen that g · Γ x · g- 1 is generated by the matrices T h = ± „ 1 h 1 « ,- I = „- 1- 1 « . The number h is also equal to the index of the subgroup g · Γ x · g- 1 in SL(2 , Z ) ∞ = ( T,- I ) . In particular, all x from the same cusp of Γ define the same number h . We shall call it the index of the cusp. Let f ( τ ) be a holomorphic function satisfying (6.1). For each x ∈ Q ∪ {∞} consider the function φ ( τ ) = f ( τ ) | k g- 1 , where g · x = ∞ for some g ∈ SL(2 , Z ). We have φ ( τ ) | k g Γ x g- 1 = f ( τ ) | k g- 1 g Γ x g- 1 = f ( τ...
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