In their paper, “Definable subgroups of algebraic groups over finite fields,” Hrushovski and Pillay write

… if V is an absolutely irreducible variety of dimension d over the finite field F_q, then the cardinality of V(F) is “roughly” q^d. So for *nonstandard* q, the cardinality of V(F) is exactly q^d.

Do I have any nonstandard readers who can explain what is meant by this provocative statement?

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One interpretation is in the context of pseudo-finite fields (infinite models of the theory of finite fields, in the sense of model theory); I think those have a “q” associated with them which is an “infinitely large” number (which can be interpreted with ultrafilters in some way).

There might be more information in the paper of Chatzidakis, van den Dries and Macintyre that they refer to.

I don’t have access to the paper, but it may boil down to how they are defining the cardinality of a nonstandard set – perhaps they are identifying any two nonstandard numbers which differ by a multiplicative factor of 1+o(1). Otherwise, I don’t see how the lower order terms will vanish in the nonstandard case; they are infinitesimal with respect to the main term, but they are not completely zero.

I wasn’t on board with this interpretation at first, because of the explicit use of the word exactly. However, reading the surrounding part of the paper, it seems like it might be right.

Without defining terms, the sentence following the one JSE writes goes:

The interpretation of the rational number mu is similar: if X has dimension d and is defined by the formula phi(x), then to say mu(X)=mu means that for any finite field F the cardinality of phi^F is roughly mu q^d.

Sounds identical to Terry’s interpretation, to me.

But if Brian and Terry are right, then this “cardinality” function isn’t additive for unions of varieties (as Brian pointed out to me by e-mail) and that makes me very resistant to calling it “cardinality” at all.

Actually, David Treumann was the one that pointed that out us.

I may be misunderstanding the factor of q^(-1/2) in the Lang-Weil estimate here, but could it be something like this? If the intended meaning is that the cardinality is also a nonstandard integer (along with q), then if the estimate has a noninteger part that is going to zero in q, that term becomes infinitesimal. Since you know that the actual cardinality is an integer, the infinitesimal has to be zero. Unfortunately, looking at the actual formula for Lang-Weil, I think this doesn’t really make sense.

Or, is there something known about the distribution of the error term that implies that in all but finitely many cases of q the square-root term is actually zero?

For the benefit of those without the paper, the … in JSE’s quote says something like:

“At the root of [the statement] are the Lang-Weil estimates …”

Now the Lang-Weil estimates say,

#V(F) = (1+o(1)) q^d,

which is what Terry said.

Since the authors don’t refer to anything stronger (like, for all but finitely many, the error term is zero), I don’t think they could have meant anything stronger than this.

Why not email the authors to ask them directly what they’re talking about?

Perhaps one might want to view the Hrushovski-Pillay cardinality not as taking values in the semiring of positive nonstandard integers, but rather in the semiring of positive nonstandard integers modulo equivalence by multiplicative (1+o(1)) factors. This is still a semiring (with a partially tropical flavour) and cardinality is still additive here. (Similarly, dimension is additive in the tropical max-plus semi-ring.)

Personally, I would have perhaps used a different term (e.g. “principal cardinality”) but this is largely a matter of taste. It could be that cardinality is mainly introduced here as a waystation to the notion of dimension (I know that dimensions of nonstandard sets are important in the type of model theory that Hrushovski and Pillay excel at).

Well, in the spirit of this comment, here’s what I heard back from Anand Pillay, who kindly let me pass this along:

—-

I am not sure what I or we meant at the time. But literally it is wrong (e.g. multiplicative group has cardinality q-1).

What is correct is that if we work over a nonstandard finite field k (in nonstandard model of set theory, with nonstandard reals etc., cardinalities of “finite” sets…), and V is an absolutely irreducible variety over k, of dimension d, then the nonstandard real number (#(V)/(#k)^d) has standard part 1, namely is within an infinitesimal nonstandard real of 1.

—–

I guess that settles the original question! But I like Terry’s interpretation. It makes one wonder: what are the interesting homomorphisms from the ring of motives K_0(Var_k) to the tropical semiring? Or to the “semiring of leading terms,” which is really the one where Hrushovski-Pillay takes values? Or to the yet more refined semiring of sequences of integers up to asymptotic equivalence, as Terry suggests?

To me, Lang-Weil means error term q^(d-1/2), since this is what the method

gives (and I think this is what they prove).