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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.