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MODULAR FORMS-page75

MODULAR FORMS-page75 - E 12 We can write θ L = 1 2 (12 E...

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71 In fact there exists only one even unimodular lattice in R 8 (up to equivalence of lattices). The lattice is the famous E 8 lattice, the root lattice of simple Lie algebra of type E 8 . Fig.2 Here the diagram describes a symmetric matrix as follows. All the diagonal el- ements are equal to 2. If we order the vertices, then the entry a ij is equal to - 1 or 0 dependent on whether the i -th vertex is connected to the j -th vertex or not, respectively. Take n = 16. Since M 4 (Γ(1)) = C E 8 , we obtain, by comparing the constant coefficients, θ L = E 8 / 2 ζ (8) . In particular, we have r L ( m ) = 16 σ 7 ( m ) /B 4 , (7.11) where B 4 is the fourth Bernoulli number (see Lecture 6). There exist two even uni- modular lattices in R 16 . One is E 8 E 8 . Another is Γ 16 defined by the following graph: Fig.3 Now let n = 24. The space M 6 (Γ(1)) is spanned by Δ and
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Unformatted text preview: E 12 . We can write θ L = 1 2 ζ (12) E 12 + c L Δ . This gives r L ( m ) = 65520 691 σ 11 ( m ) + c L τ ( m ) , (7.12) where τ ( m ) is the Ramanujan function (the coefficient at q m in Δ). Setting m = 1, we get c L = r L (1)-65520 691 . (7.13) Clearly, c L 6 = 0. Except obvious examples E 8 ⊕ E 8 ⊕ E 8 or E 8 ⊕ Γ 16 there are 22 more even uni-modular lattices of rank 24. One of them is the Leech lattice Λ. It differs from any other lattice by the property that r Λ (1) = 0. So, r L ( m ) = 65520 691 ( σ 11 ( m )-τ ( m )) , (7.14) In particular, we see that τ ( m ) ≡ σ 11 ( m ) mod 691 . This is one of the numerous congruences satisfied by the Ramanujan function τ ( m )....
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