Let $C$ be smooth projective curve defined over a finite field $\mathbb{F}_q$. Let $$Z(C,u)=\exp(\sum_{n \ge 1} N_r(C) u^r/r) \in \mathbb{Z}[[u]]$$ be its zeta function, where $N_r$ is the number of $\mathbb{F}_{q^r}$-points on $C$. It is well known (Weil) that $$Z(C,u)=\frac{P_C(u)}{(1-qu)(1-u)},$$ where $P_C$ is a polynomial of degree twice the genus of $C$, that satisfies $P_C(0)=1$ and $P_C(\alpha)=0\implies |\alpha|=q^{-1/2}$.

What is the standard name for curves $C$ such that the roots of $P_C$ are roots of unity times $q^{-1/2}$? And what is a suitable reference where this name is used?

Also,

What is the motivation behind studying such curves?

I have seen that sometimes such curves are called `supersingular'. I have also seen many other definitions for supersingular curves, not involving the roots of $P_C$. But I have never seen how (and if...) these different definitions are related, which leaves me a bit confused. I list these definitions below.

- Here van der Geer and van der Vlugt define supersingular curves as curves whose jacobians are isogenous to a product of supersingular elliptic curves (over $\overline{\mathbb{F}_q}$).
- Wikipedia describes a definition involving Newton polygons.
- Rosen (PDF) defines supersingular curves using class numbers.
- Here Kodama and Washio mention a definition involving the Hasse-Witt invariant, and also relate this invariant to the roots of $P_C$.