I was reading a paper and they mentioned the Rosati form. Particularly, what they said was:


Let $A$ be an abelian surface defined over $k$ such that $ST\_A^0$ (the connected component of the Sato-Tate group) $ = U(1) \times U(1)$. Then the matrices in $M\_4(\mathbb{C})$ commuting with $U(1) \times U(1)$ are


$$\Bigg\{\begin{pmatrix}a&0&0&0\\0&b&0&0\\0&0&c&0\\0&0&0&d\end{pmatrix}: a,b,c,d\in\mathbb{C}\Bigg\}$$


and the $\textit{Rosati form}$ is a scalar multiple of $2ab+2cd = \frac{1}{2}((a+b)^2 - (a-b)^2 + (c+d)^2 - (c-d)^2)$. 


I have done a Google search and can find nothing that references a Rosati form. I have found references to a Rosati involution. Is this the same thing? If not where could I find a definition of the Rosati form>



What is Rosati Form
the Rosati form is defined in definition 2.18 of [*Sato-Tate distributions and Galois endomorphism modules in genus 2*](http://arxiv.org/abs/1110.6638)


![](https://ilorentz.org/beenakker/MO/rosati.png)


