Lectures # 7: The Class Number Formula For Positive Definite Binary Quadratic Forms

Notice that this means if we change variables ( x′ y′ ) = v′ = Av (the entries of A are integers) then Q(x′, y′) = (Av) QAv = v (A QA)v. Therefore, if Q represents a number then so does A QA. In particular, if A has an inverse with integer entries, then we get that Q and A QA represent all the same numbers. Clearly if A has an inverse B with integer entries, then det A detB = det AB = 1, thus det A = ±1. Furthermore one can easily show that if det A = ±1 then A has an integer inverse. Lagrange defined two forms Q and Q′ to be equivalent if there exists A with determinant ±1 such that A QA = Q′. Gauss strengthened this notion as follows. Definition 1.5. We say that two forms Q and Q′ are properly equivalent if there exists a matrix A such that A QA = Q′ and det A = 1. If there exists a matrix A such that A QA = Q′ and detA = −1 then we call the two BQFs improperly equivalent. Unless otherwise noted, when we say equivalent we mean properly equivalent.