Unformatted text preview: are properly equivalent if they diﬀer by a substitution with determinant equal to 1. In other words, we restrict ourselves with with the subgroup SL(2 , Z ) of GL(2 , Z ). Since GL(2 , Z ) = SL(2 , Z ) ∪ ± 11 ² SL(2 , Z ) and ± 11 ² ( ax 2 + 2 bxy + cy 2 ) = ax 22 bxy + cy 2 we obtain that each f is properly equivalent to a form ax 2 + 2 bxy + cy 2 , where ( a,b,c ) ∈ ¯ Ω and ¯ Ω = { ( a,b,c ) ∈ R 3 :  2 b  ≤ c ≤ a,a,acb 2 > } . Deﬁnition. We shall say that f = ax 2 + 2 bxy + cy 2 is properly reduced if ( a,b,c ) ∈ ¯ Ω....
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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.
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