# zero map of hermitian forms

$$n \times n$$ matrix $$A$$ with complex entries is called Hermitian if $$A^{*}=A,$$ where $$A^{*}=\bar{A}^{T}$$

$$H(\Bbb{C}):=\left\{A \in M_{2}(\Bbb{C}) \mid A^{*}=A\right\}$$ $$H(\Bbb{C})$$ consists of 2 by 2 the matrices $$A=\left(\begin{array}{ll} a & b \\ \bar{b} & d \end{array}\right)$$ where $$a, d \in \Bbb{R}$$ and $$b \in \Bbb{C}$$. If $$A \in H(\Bbb{C})$$ then the associated binary Hermitian form is the semi quadratic map $$Q: \mathbb{C} \times \mathbb{C} \rightarrow \Bbb{R}$$ defined by $$Q(X, Z)=(X, Z)\left(\begin{array}{c} a & b \\ \bar{b} & d \end{array}\right)(X, Z)^{*}=a|X|^2+b X Z+\bar{b} \bar{X} Z+d|Z|^2$$

Let $$H^{+}(\mathbb{C})$$ denote the set of positive definite binary quadratic hermitian forms and $$\Delta(Q):=\operatorname{det}(Q)=a d-|b|^{2}$$

Definition: Let $$\mathcal{H}_3=\Bbb{C}\times (0,\infty)$$ be the upper half space. The map $$\xi: H^{+}(\mathbb{C}) \rightarrow \mathcal{H}_{3}$$ defined by $$\xi\begin{pmatrix}a & b \\ \overline{b} & d \end{pmatrix}=\frac{b}{a}+\frac{\sqrt{\Delta(Q)}}{a} \cdot j$$ is called the "zero map" for binary quadratic Hermitian forms.

My question: Why is it called zero map? I expect that $$Q(X,1)=a|X|^2+bX+\overline{bX}+d$$ admits $$\frac{b}{a}+\frac{\sqrt{\Delta(Q)}}{a} j$$ as a zero since the zero of an positive definite binary quadratic form $$f(x,y)=ax^2+bxy+cy^2$$ is defined as $$\frac{-b+\sqrt{\Delta}}{2a}$$, which is really a zero of $$f(x,1)$$. But when I set $$X=\frac{b}{a}+\frac{\sqrt{\Delta(Q)}}{a} j$$, I get $$Q(X,1)=\frac{a^2+b^2+\overline{b^2}+d^2}{d}$$ using the algebraic properties of quaternions. Do I have wrong expectation?

No, it isn't a quaternionic zero of $$H(X,1)=0$$. For instance (and I think you can reduce to this case with a linear change of variable) consider $$H(X,Z)=|X|^2+|Z|^2$$. Clearly $$|X|^2=-1$$ has no solutions in $$\mathbb{H}$$.
More important is the equivariance of the $$PSL(2,\mathbb{C})$$-action on forms/zero sets. Take a look at Elstrodt, Grunewald, Mennicke, Groups Acting on Hyperbolic Space for more on the subject.