MODULAR FORMS-page63

MODULAR FORMS-page63 - 59 Setting t = e2iz , we rewrite the...

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59 Setting t = e 2 πiz , we rewrite the left-hand side as follows: π cot( πz ) = π cos πz sin πz = e πiz + e - πiz e πiz - e - πiz ) = t + 1 t - 1 = (1 - 2 X m =0 t m ) . Differentiating k - 1 2 times in z , we get ( k - 1)! X m Z ( z + m ) - k = (2 πi ) k X m =1 m k - 1 t m . This gives us the needed Fourier expansion of E k ( τ ). Replace in above z with , set q = e 2 πiτ to obtain E k ( τ ) = 2 ζ ( k ) + X n N 2(2 πi ) k ( k - 1)! ` X m =1 m k - 1 q nm ´ . (6.25) It is obviously convergent at q = 0. So, we obtain that E k ( τ ) is a modular form of weight k/ 2 with respect to the full modular group Γ(1) . It is called the Eisenstein form of weight k/ 2. Recall that k must be even and also k 4. One can rewrite (6.21) in the form E k ( τ ) = 2 ζ ( k ) + 2(2 πi ) k ( k - 1)! X m =1 σ k - 1 ( m ) q m , (6.26) where σ n ( m ) = X d | m d n = sum of n th powers of all positive divisors of m. Now we observe that we have 3 modular forms of weight 6 with respect to Γ(1). They are g 3 2 = 60 3 E 3 4 ,g 3 = (140) 2 E 2 6 , Δ. There is a linear relation between these 3 forms: Theorem 6.1. (2 π ) 12 Δ = g 3 2 - 27 g 2 3 . Proof. First notice that g 3 2 - 27 g 2 3 is equal to the discriminant of the cubic polynomial
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