It is known that the Dirichlet composition of two binary quadratic forms induces an abelian group structure on the set of positive definite quadratic forms with a given discriminant. What is that group structure? Given any two quadratic forms and the Dirichlet composition formulæ, what is the group operation under which it forms an abelian group?
2026-02-27 23:45:43.1772235943
Group structure on the Dirichlet binary composition of quadratic forms
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The identity element is the "principal" form, meaning the form that represents $1.$ This is (the class of) $\langle 1,b,c \rangle,$ same as the class of $\langle c,b,1 \rangle.$ The sameness boils down to the fact that $b$ is divisible by $1.$
Given a form with coefficients $\langle a,b,c \rangle,$ the inverse in the group is $\langle c,b,a \rangle.$ The traditional word for this class is "opposite." Another representative for the opposite class is $\langle a,-b,c \rangle.$
In both cases "form" is taken to mean the $SL_2 \mathbb Z$ equivalence classes.