MODULAR FORMS-page79

MODULAR FORMS-page79 - is bounded on H(it is not true if f...

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75 (i) Show that r L C (2) = 48 + 16 A 4 , r L C (4) = 2 8 A 8 + 640 A 4 + 1104 . (ii) Using (7.13) show that A 8 = 759 - 4 A 4 . 7.9 Let A = n = -∞ A n be a commutative graded algebra over a field F . Assume A has no zero divisors, A 0 = F · 1 and dim A N > 1 for some nN > 0. Show that A n = 0 for n < 0. Apply this to give another proof that M k (Γ) = 0 for k < 0. 7.10 Find an explicit linear relation between the modular forms E 16 ,E 2 8 and E 4 E 10 , where E 2 k denotes the Eisenstein series. Translate this relation into a relation between the values of the functions σ d ( m ). 7.11 Let f ( τ ) be a parabolic modular form of weight k with respect to Γ(1). (i) Show that the function φ ( τ ) = | f ( τ ) | (Im τ ) k is invariant with respect to Γ(1). (ii) Show that φ ( τ
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Unformatted text preview: ) is bounded on H (it is not true if f is not cuspidal). (iii) Show that the coefficient a n in the Fourier expansion f ( τ ) = P a n q n can be computed as the integral a n = Z 1 f ( x + iy ) e-2 πin ( x + iy ) dx. (iv) Using (iii) prove that | a n | = O ( n k ) (Hecke’s Theorem). 7.12 Let L be an even unimodular lattice in R 8 k and r L ( m ) be defined as in Example 7.2. Using the previous exercise show that r L ( m ) = 8 k B 2 k σ 4 k-1 ( m ) + O ( m 2 k ) . 7.13 Let L = E 8 ⊕ E 8 ⊕ E 8 . Show that θ L = 1 ζ (12) E 12 + 432000 691 Δ ....
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