For varieties over a field of characteristic $p$, I saw people talking about supersingular varieties.
I wanted to ask "why are supersingular varieties interesting". However, as I don't want to ask an MO question with no background/context I thought I'd better define what a supersingular variety is. Unfortunately I can't. (And search engine doesn't help…) Can anyone here help me with this and explain why they are interesting?
A little bit more context: I have seen the definition of supersingular elliptic curves on textbooks by Hartshorne and Silverman. When I read about abelian varieties I saw "for abelian varieties of $\text{dim}>2$ being supersingular $\neq p$ rank being 0".
Illusie mentioned (in the "Motives" volume) that for $X$ a variety over a perfect field of characteristic $p$, $X$ is said to be ordinary if "$H_\text{cris}^*(X/W(k))$ has no torsion and $\text{Newt}_m(X)=\text{Hdg}_m(X)$ for all $m$". My guess is being supersingular should correspond to the other extreme, but what precisely is it? ($\text{Newt}_m(X)$ being a straight line for all $m$?)
Best Answer
Not an answer, but some historical context (which I think is correct). An elliptic curve over $\mathbb{C}$ used to be called "singular" if its endomorphism ring was larger than $\mathbb{Z}$, i.e., what we now call having complex multiplication. Presumably this use of the word singular was to indicate that the curve was unusual. Then, when people looked at elliptic curves over finite fields, the found that some of them had endomorphism rings that were even larger than an order in a quadratic imaginary field, so those curves were "supersingular" in the sense of being even more unusual. Of course, it turns out that an alternative way to characterize those curves is as having no $p$-torsion (over the algebraic closure of their base field).