MODULAR FORMS-page78

MODULAR FORMS-page78 - 74 LECTURE 7. THE ALGEBRA OF MODULAR...

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74 LECTURE 7. THE ALGEBRA OF MODULAR FORMS (i) Show that L C is an integral lattice if and only if for any x = ( ± 1 ,...,± n ) C the number wt( x ) = # { i : ± i 6 = 0 } (called the weight of x ) is divisible by 4. In this case we say that C is a doubly even linear code. (ii) Show that the discriminant of the lattice L C is equal to 2 n - 2 k , where k = dim C . (iii) Let C = { y F n 2 : x · y = 0 , x C } . Show that L C is integral if and only if C C . (iv) Assume C is doubly even. Show that L C = L * C . In particular, L C is a uni- modular even lattice if and only if C = C (in this case C is called a self-dual code ). (v) Let C F n 2 be a self-dual doubly even code. Show that n must be divisible by 8. 7.6 Let A ( τ ) = ϑ (0; τ ) ,B ( τ ) = ϑ 1 2 0 (0; τ ). (i) Show that ( A ( - 1 ) ,B ( - 1 )) = ( τ i ) 1 / 2 ( A ( τ ) ,B ( τ )) · 1 2 1 2 1 2 - 1 2 ! . (ii) Show that the expression
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