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MODULAR FORMS-page73 - L is an integral lattice in R n If v...

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69 7.3 Let us give some examples. Example 7.1 . We know that the Eisenstein series E 2 k is a modular form of weight k with respect to Γ(1). Since M 4 (Γ(1)) = C g 2 2 = C E 2 4 , comparing the constant coefficients in the Fourier expansions we obtain E 8 = ζ (8) 2 ζ (4) 2 E 2 4 . Comparing the other coefficients we get a lot of identities between the numbers σ k ( n ). For example, we have σ 7 ( n ) = σ 3 ( n ) + 120 X 0 <m<n σ 3 ( m ) σ 3 ( n - m ) . (7.5) Similarly we have E 10 = ζ (10) 2 ζ (4) ζ (6) E 4 E 6 . This gives us more identities. By the way our old relation (2 π ) 12 Δ = g 3 2 - 27 g 2 3 gives the expression of the Ramanujan function τ ( n ) defined by Δ = q Y m =1 (1 - q m ) 24 = X n =0 τ ( n ) q n in terms of the functions σ k ( n ): τ ( n ) = 65 756 σ 11 ( n ) + 691 756 σ 5 ( n ) - 691 3 X 0 <m<n σ 5 ( m ) σ 5 ( n - m ) . (7.6) We shall prove in Lecture 11 that τ ( n ) satisfies τ ( nm ) = τ ( n ) τ ( m ) if ( n, m ) = 1 , τ ( p k +1 ) = τ ( p ) τ ( p k ) - p 11 τ ( p k - 1 ) if p is prime, k 0 . Example 7.2 . Let L be a lattice in R n of rank n such that for any v L the Euclidean norm || v || 2 takes integer values. We say that
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Unformatted text preview: L is an integral lattice in R n . If ( v 1 ,...,v n ) is a basis of Λ, then the dot products a ij = v i · v j define an integral symmetric non-degenerate matrix, hence an integral quadratic form Q = n X i,j =1 a ij x i x j . Obviously for any v = ( a 1 ,...,a n ) 6 = 0 we have Q ( v ) = || v || 2 > . In other words, Q is positive definite. Conversely given any positive definite integral quadratic form Q as above, we can find a basis ( e 1 ,...,e n ) such that Q diagonalizes, i.e. its matrix with respect to this basis is the identity matrix. Let φ : R n → R n be the linear automorphism which sends the standard basis ( e 1 ,...,e n ) to the basis...
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