TITLE: Elkies' supersingularity theorem in higher dimension
QUESTION [19 upvotes]: The following is a theorem of Elkies:
Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero.
Allen Knutson and I would like a similar theorem for higher dimensional Calabi-Yau varieties. Unfortunately, we've been told that this is probably open. (References for the fact that it is open are appreciated.) But we don't need the full strength of Elkies' result. It would be enough for us to know the following:
Let $X$ be an $n$-dimensional, smooth, complete Calabi-Yau variety over $\mathbb{Q}$, for $n>0$. Write $X/p$ for the fiber of $X$ over $p$. Let $T(p)$ be the action of Frobenius on $H^n(\mathcal{O}, X/p)$. Are there infinitely many $p$ for which $T(p) \neq 1$? Also, in the same generality, are there infinitely many primes for which $T(p) \neq 0$?

REPLY [5 votes]: Your question is connected with Frobenius splitting. Namely, we have the following fact (Brion-Kumar "Frobenius splitting methods in geometry and representation theory", Remark 1.3.9 (ii)):

A complete smooth variety $X$ over an algebraically closed field of characteristic $p>0$ is Frobenius split if and only if the pullback map $F^*: H^n(X, \omega_X)\to H^n(X, \omega_X^p)$ is nonzero.

Since in your case $\omega_X = \mathcal{O}_X$, this is exactly the same question. We can be even more general and ask questions like "which characteristic zero varieties are (or are not) Frobenius split for $p$ sufficiently large (or infinitely many $p$)". 
One result of this kind (Brion-Kumar, exercise in paragraph 1.6) is that if $X$ is Fano, then $X/p$ is Frobenius split for $p$ large enough.
Moreover, in their recent work, Mustata and Smith propose a conjecture of this kind in the local case. I don't know enough details to state it precisely, but it says something like "$fpt(X/p) = lct(X)$ for infinitely many $p$", where $X$ is a singularity we reduce mod $p$, $fpt$ is the Frobenius pure threshold and $lct$ is the log-canonical threshold. I heard that this conjecture will imply that all characteristic zero abelian varieties become ordinary when reduced modulo infinitely many $p$.