MODULAR FORMS-page8

MODULAR FORMS-page8 - R 3 : 0 2 b a c,a > ,ac...

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4 LECTURE 1. BINARY QUADRATIC FORMS among all vectors not equal to ± v 0 . I claim that ( v 0 , w 0 ) forms a basis of Λ. Assume it is false. Then there exists a vector x Λ such that x = a v 0 + b w 0 where one of the coefficients a,b is a real number but not an integer. After adding some integral linear combination of v 0 , w 0 we can assume that | a | , | b | ≤ 1 2 . If a,b 6 = 0, this gives k x k 2 = | a | 2 k v 0 k 2 + | b | 2 k w 0 k 2 + 2 ab v 0 · w 0 < ( | a |k v 0 k + | b |k w 0 k ) 2 1 2 k w 0 k 2 contradicting the choice of w 0 . Here we have used the Cauchy inequality together with the fact that the vectors v 0 and w 0 are not proportional. If a or b is zero, we get k x k = 1 2 k v 0 k or k x k = 1 2 k w 0 k , again a contradiction. Now let us look at g . The square of the two diagonals d ± of the parallelogram formed by the vectors v 0 , w 0 is equal to d 2 ± = k v 0 k 2 ± 2 v 0 · w 0 + k w 0 k 2 . Clearly d ± ≥ k w 0 k . By construction, k w 0 k ≥ k v 0 k . Thus 2 | v 0 · w 0 | ≤ k v 0 k 2 k w 0 k 2 . It remains to change v 0 to - v 0 , if needed, to assume that B = v · w 0 0. Definition. A positive definite binary quadratic form ax 2 +2 bxy + cy 2 is called reduced if 0 2 b a c. The previous theorem says that each positive definite binary quadratic form is equivalent to a reduced form. Let Ω = { ( a,b,c )
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Unformatted text preview: R 3 : 0 2 b a c,a &gt; ,ac &gt; b 2 } . (1.3) By Theorem 1.2, any positive denite binary quadratic form is equivalent to a form ax 2 + 2 bxy + cy 2 , where ( a,b,c ) . 1.3 Let us nd when two reduced forms are equivalent. To do this we should look at the domain from a dierent angle. Each positive denite quadratic form f = ax 2 + 2 bxy + cy 2 can be factored over C into product of linear forms: f = ax 2 + 2 bxy + cy 2 = a ( x-zy )( x- zy ) , where z =-b a + i ac-b 2 a . (1.4) It is clear that f is completely determined by the coecient a and the root z . Observe that Im z &gt; . We have a bijective correspondence f = ax 2 + 2 bxy + cy 2 ( a,z ) from the set of positive denite binary quadratic forms to the set R + H , where H = { z C : Im z &gt; }...
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