2 LECTURE 1. BINARY QUADRATIC FORMS Let x = m 1 v + m 2 w ∈ Λ. The length of x is given by the formula k x || 2 = || m 1 v + m 2 w || 2 = ( m 1 ,m 2 ) ± v · v v · w v · w w · w ²± m 1 m 2 ² = am 2 1 + 2 bm 1 m 2 + cm 2 2 , where a = v · v , b = v · w , c = w · w . (1.1) Let us consider the (binary) quadratic form (the distance quadratic form of Λ) f = ax 2 + 2 bxy + cy 2 . Notice that its discriminant satisﬁes D = 4( b 2-ac ) =-4 A ( v , w ) 2 <0 . (1.2) Thus f is positive deﬁnite. Given a positive integer n one may ask about integral solutions of the equation f ( x,y ) = n. If there is an integral solution ( m 1 ,m 2 ) of this equation, we say that the binary form f represents the number n . Geometrically this means that the circle of radius √ n centered at the origin contains one of the points x = m 1 v + m 2 w of the lattice Λ. Notice that the solution of this problem depends only on the lattice Λ but not on the form
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This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.