In his answer to this question, Scott Carnahan mentions "mirror symmetry mod p". What is that?

(Some kind of) Gromov-Witten invariants can be defined for varieties over fields other than $\mathbb{C}$. Moreover other things that come up in mirror symmetry, like variation of Hodge structure, and derived categories of coherent sheaves, also make sense. (Though I can't imagine that it's possible to talk about Fukaya categories...) Can we formulate any sort of sensible mirror symmetry statement, similar to say that of Candelas-de la Ossa-Green-Parkes relating Gromov-Witten invariants of a quintic threefold to variation of Hodge structure of the mirror variety, when the varieties are over some field other than $\mathbb{C}$? In particular, can we do anything like this for fields of positive characteristic?

I googled "arithmetic mirror symmetry" and "mirror symmetry mod p", and I found some stuff about the relationship between the arithmetic of mirror varieties, but nothing about Gromov-Witten invariants. I did find notes from the Candelas lectures that Scott referred to, but I wasn't able to figure out what was going on in them.

More generally, there are many examples of mathematical statements about complex algebraic varieties which come from physics/quantum field theory/string theory. Some of these statements (maybe with some modification) can still make sense if we replace "variety over $\mathbb{C}$ with "variety over $k$", where $k$ is some arbitrary field, or a field of positive characteristic, or whatever. Are there any such statements which have been proven?

Edit: I'm getting some answers, and they are all sound very interesting, but I'm still especially curious about whether anybody has done anything regarding Gromov-Witten invariants over fields other than $\mathbb{C}$.

notasking about p-adic physics. But I welcome someone else to ask a question about it... – Kevin H. Lin Jan 28 '10 at 3:44