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Unformatted text preview: E 12 . We can write L = 1 2 (12) E 12 + c L . This gives r L ( m ) = 65520 691 11 ( m ) + c L ( m ) , (7.12) where ( m ) is the Ramanujan function (the coecient at q m in ). 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 16 there are 22 more even unimodular lattices of rank 24. One of them is the Leech lattice . It diers from any other lattice by the property that r (1) = 0. So, r L ( m ) = 65520 691 ( 11 ( m ) ( m )) , (7.14) In particular, we see that ( m ) 11 ( m ) mod 691 . This is one of the numerous congruences satised by the Ramanujan function ( m )....
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 Fall '09
 ONTONKONG
 Algebra

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