Let $A$ be an abelian surface over an finite field $\mathbb{F}_q$. In particular, I am interested in the case when $A$ is a Jacobian variety. Is the Kummer surface $K_A/\mathbb{F}_q$ Shioda-supersingular (that is the Picard number $\rho(K_A/\mathbb{F}_q)$ is equal to the second $l$-adic Betti number $b_2(K_A) = 22$) iff $\mathbb{F}_q$-unirational?
This is true over an algebraically closed field of a characteristics $p > 3$. (http://link.springer.com/article/10.1007/BF01350715, Corollary 2; http://link.springer.com/article/10.1007/s00222-014-0547-7).
In my opinion, this question is interesting, because exceptions can be over non-closed fields. For example, $\rho(K_A) \neq 21$ over algebraically closed fields (http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1974_4_7_4/ASENS_1974_4_7_4_543_0/ASENS_1974_4_7_4_543_0.pdf, 3 page), but this is wrong over finite fields (http://arxiv.org/abs/1105.4993).
Proposition 8.3 of http://reh.math.uni-duesseldorf.de/~schroeer/publications_pdf/kummer.pdf states the equality $\rho(K_A/\mathbb{F}_q) = 21$ for $\mathbb{F}_q$-unirational Kummer surfaces $K_A$, where $\mu_3 \not\subset \mathbb{F}_q$, but this article doesn't state existence of such $K_A$ over such fields if I'm not mistaken.
