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MODULAR FORMS-page70

# MODULAR FORMS-page70 - has a zero or pole a at the boundary...

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66 LECTURE 7. THE ALGEBRA OF MODULAR FORMS Note that when f is a modular form with respect to a group Γ we have ν g · τ ( f ) = ν τ ( f ) , g Γ . For each τ ∈ H let m τ = 8 > < > : 2 if τ Γ(1) · i , 3 if τ Γ(1) · e 2 πi/ 3 , 1 otherwise. (7.3) Lemma 7.1. Let f ( τ ) be a modular form of weight k with respect to the full modular group Γ(1) . Then X τ ∈H / Γ(1) ν τ ( f ) m τ = k 6 . Proof. Consider the subset P of the modular ﬁgure D obtained as follows. First delete the part of D deﬁned by the condition Im τ > h for suﬃciently large h such that f has no zeroes or poles for Im τ h . Let C r ( ρ ) ,C r ( ρ 2 ) ,C r ( i ) be a small circle of radius r centered at ρ = e πi/ 3 at ρ 2 and at i , respectively. Delete from D the intersection with each of these circles. Finally if f ( z
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Unformatted text preview: ) has a zero or pole a at the boundary of D we delete from D its intersection with a small circle of radius r with center at a . Fig.1 Applying the Cauchy Residue Theorem we obtain 1 2 πi Z ∂P f dτ f = X τ ∈ P ν τ ( f ) = X τ ∈ P ν τ ( f ) m τ . When we integrate over the part ∂P 1 of the boundary deﬁned by Im τ = h we obtain 1 2 πi Z ∂P 1 f dz f =-ν ∞ ( f ) . In fact, considering the Fourier expansions of f at ∞ , we get f ( τ ) = ∞ X n = ν ∞ ( f ) a n e 2 πinτ , f ( τ ) = ∞ X n = ν ∞ ( f ) (2 πin ) a n e 2 πniτ ....
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