Supersingularity and Roots of Unity of Zeta Functions

Question

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 $P_C$.

Answer 1

The names supersingular and superspecial appear in the literature in not a very consistent way.

Passing from $\mathbb{F}_q$ to $\mathbb{F}_{q^n}$ changes the roots $\alpha$ to $\alpha^n$. If the roots are roots of unity times $q^{-1/2}$, then passing to a suitable extension, we can assume that they are equal to $q^{-1/2}$ and that $q$ is a square. In this case $P_C(t)= (1-q^{1/2}t)^{2g}$ and the fact that this condition is equivalent to the Jacobian being isogenous to $E^g$, $E$ supersingular with $P_E(t)= (1-q^{1/2}t)^{2}$ follows from Tate's isogeny theorem.

The class number is $P_C(1) = (1-q^{1/2})^{2g}$, which is smallest it can be. Also the number of points $\# C(\mathbb{F}_q) = q + 1 -2gq^{1/2}$ is the smallest it can be. This is a major source of interest in these curves. Also, commonly studied are the maximal curves that attain the Weil bound $\# C(\mathbb{F}_q) = q + 1 +2gq^{1/2}$ and so are as above after extending to the quadratic extension.

The Hasse invariant of these curves is zero because of Manin's theorem relating the characteristic polynomial of the Cartier operator with the reduction modulo the characteristic $p$ of $P_C$. I don't think the converse holds.