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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 . I claim that ( v , w ) forms a basis of Λ. Assume it is false. Then there exists a vector x Λ such that x = a v + b w where one of the coefficients a, b is a real number but not an integer. After adding some integral linear combination of v , w we can assume that | a | , | b | ≤ 1 2 . If a, b = 0, this gives x 2 = | a | 2 v 2 + | b | 2 w 2 + 2 ab v · w < ( | a | v + | b | w ) 2 1 2 w 2 contradicting the choice of w . Here we have used the Cauchy inequality together with the fact that the vectors v and w are not proportional. If a or b is zero, we get x = 1 2 v or x = 1 2 w , again a contradiction. Now let us look at g . The square of the two diagonals d ± of the parallelogram formed by the vectors v , w is equal to d 2 ± = v 2 ± 2 v · w + w 2 . Clearly d ± w . By construction, w v . Thus 2 | v · w | ≤ v 2 w 2 . It remains to change v to - v , if needed, to assume that B = v · w 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.
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Unformatted text preview: R 3 : 0 ≤ 2 b ≤ a ≤ c,a > ,ac > b 2 } . (1.3) By Theorem 1.2, any positive definite binary quadratic form is equivalent to a form ax 2 + 2 bxy + cy 2 , where ( a,b,c ) ∈ Ω. 1.3 Let us find when two reduced forms are equivalent. To do this we should look at the domain Ω from a different angle. Each positive definite 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 coefficient a and the root z . Observe that Im z > . We have a bijective correspondence f = ax 2 + 2 bxy + cy 2 → ( a,z ) from the set of positive definite binary quadratic forms to the set R + ×H , where H = { z ∈ C : Im z > }...
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