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Unformatted text preview: 8430 HANDOUT 3: ELEMENTARY THEORY OF QUADRATIC FORMS PETE L. CLARK 1. Basic definitions An integral binary quadratic form is just a polynomial f = ax 2 + bxy + cy 2 with a,b,c Z . We define the discriminant of such a form to be ( f ) = b 2- 4 ac . A quadratic form is nondegenerate if ( f ) 6 = 0. Generally we restrict atten- tion to nondegenerate forms. We say a quadratic form q ( x,y ) represents an integer N if there exist integers x and y with q ( x,y ) = N . We say that q primitively represents 1 N if there exist relatively prime integers x and y such that q ( x,y ) = N . Let f = ax 2 + bxy + cy 2 be a quadratic form and suppose that d = gcd( a,b,c ) > 1. Then f can only represent integers which are multiples of d in particular it can represent at most one prime, and no primes at all if d is not prime. Moreover f represents dN iff the integral form 1 d f represents N , so there is no loss of generality in considering only the case gcd( a,b,c ) = 1. Such forms are said to be primitive . Exercise 3.1.1: Show that any quadratic form of squarefree dscriminant is primitive. For any quadratic form f , ( f ) = b 2- 4 ac b 2 (mod 4), i.e., ( f ) is 0 or 1 mod 4. Conversely, if D 0 (mod 4), then q D = x 2- D 4 y 2 is a form of discriminant D , and if D 1 (mod 4) then q D = x 2 + xy + 1- D 4 y 2 is a form of discriminant D . We have seen these forms before, of course; (1) is the norm form of the quadratic order Z [ D ] and (2) is the norm form of the quadratic order Z [ 1+ D 2 ]. Let us call the form q D the principal form of discriminant D . It has the following nice property: Proposition 1. For any integer D , 1 (mod 4) , the set of nonzero integers represented by the principal form q D is a submonoid of the multiplicative monoid 1 Coxs terminology is properly represents. In the first draft of these notes and in my first lecture on the subject I used this terminology, but because primitive representation makes much more intuitive sense, I slipped into that terminology without noticing. 1 2 PETE L. CLARK of all nonzero integers. More plainly, q D represents 1 , and if q D represents N 1 and N 2 then it represents N 1 N 2 . Exercise 3.1.2: Prove Proposition 1. 2. Matrix Representations To the binary quadratic form f ( x,y ) = ax 2 + bxy + cy 2 we may associate the 2 2 symmetric matrix M f = a b 2 b 2 c , the point being f ( x,y ) = x y a b 2 b 2 c x y . Notice that M f M 2 ( Q ), i.e., it has rational coefficients, and it it lies in M 2 ( Z ) iff b is even. A quadratic form such that M f M 2 ( Z ) a form ax 2 + 2 bxy + cy 2 is called classically integral or integer-matrix . When we wish to emphasize the slight additional generality of our definition, we refer to a form ax 2 + bxy + cy 2 (with b possibly odd) as integer valued . The following exercise justifies this terminology....
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- Fall '11