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MODULAR FORMS-page7 - 3 and hence a c b d = ab cd This can...

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3 and hence a b c d = α γ β δ a b c d α β γ δ . This can be also expressed by saying that the form f is obtained from the form f by using the change of variables x αx + βy, y γx + δy. We write this in the form f = Mf. According to Lagrange two binary quadratic forms f and g are called equivalent if one transforms to another under the change of variables as above defined by an integral matrix with determinant ± 1. An equivalence class is called the class of forms . Obviously, for any n Z , the set of integral solutions of the equations f ( x, y ) = n depends only on the class of forms to which f belongs. Also it is clear that two equivalent forms have the same discriminant. 1.2 As we saw before any lattice Λ determines a class of forms expressing the distance from a point in Λ to the origin. Conversely, given a positive definite binary form f = ax 2 + 2 bxy + cy 2 we can find a lattice Λ corresponding to this form. To do this we choose any vector v of length a and let w be the
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