Composition of Forms

Abstract

We are now ready to compare forms and groups of discriminants Δ and Δr². In the language of algebraic number theory, this is a comparison of the group of classes of ideals in the ring of integers with the group of classes of ideals in the order of index r, We recall that if

$$ R = \left( {\begin{array}{*{20}{c}} \alpha &\beta \\ \gamma &\delta \end{array}} \right) $$

is any 2 × 2 matrix with integer coefficients and determinant r, then the change of variables (1.1) takes a form f = (a, b, c) of discriminant Δ to a form

$$ {f^1} = ({a^1},{\rm{ }}{b^1}{\rm{ }}{{\rm{c}}^1}) = (a{\alpha ^2} + b\alpha {\gamma ^2} + c{\gamma ^2},{\rm{ b(}}\alpha \delta {\rm{ + }}\beta \gamma {\rm{) + 2(a}}\alpha \beta {\rm{ + c}}\gamma \delta {\rm{),a}}{\beta ^2} + b\beta \delta + c{\delta ^2}) $$

of discriminant Δr². In matrix notation this is

$$ {f^1} = ({a^1},{\rm{ }}{b^1}{\rm{ }}{{\rm{c}}^1}) = (a{\alpha ^2} + b\alpha {\gamma ^2} + c{\gamma ^2},{\rm{ b(}}\alpha \delta {\rm{ + }}\beta \gamma {\rm{) + 2(a}}\alpha \beta {\rm{ + c}}\gamma \delta {\rm{),a}}{\beta ^2} + b\beta \delta + c{\delta ^2}) $$

$$ \left( {\begin{array}{*{20}{c}} {{a^1}}&{{b^1}/2}\\ {{b^1}/2}&c \end{array}} \right) = \left( {\begin{array}{*{20}{c}} \alpha &\gamma \\ \beta &\delta \end{array}} \right)\left( {\begin{array}{*{20}{c}} a&{b/2}\\ {b/2}&c \end{array}} \right)\left( {\begin{array}{*{20}{c}} \alpha &\beta \\ \gamma &\delta \end{array}} \right), $$

which we will write as f¹ = R^TfR for brevity. We shall call such a matrix R a transformation of determinant r and shall say that f¹ is derived from f by the transformation of determinant r.