ACFA

A difference field is a field $$K$$ with an endomorphism $$\sigma$$. The theory of difference fields is inductive, and ACFA is its model companion. Therefore, models of ACFA are existentially closed difference fields.

ACFA can be explicitly axiomatized as follows. $$(K,\sigma) \models ACFA$$ if and only if the following conditions hold:


 * $$K$$ is an algebraically closed field
 * $$\sigma$$ is an automorphism of $$K$$, in particular, it is surjective
 * If $$V$$ is an irreducible affine variety over $$K$$, and if $$V^{\sigma}$$ is the image of $$V$$ under $$\sigma$$, and if $$S$$ is an irreducible closed subvariety of $$V \times V^\sigma$$ such that the projection maps $$S \to V$$ and $$S \to V^\sigma$$ are dominant, and if $$T$$ is a proper closed subvariety of $$S$$, then there is some $$v \in V$$ such that $$(v,\sigma(v)) \in S \setminus T$$.

The fact that this third condition is first-order is a little non-trivial, and uses the fact that irreducibility is definable in ACF.

ACFA is supersimple, but not stable. It is probably one of the most important examples of a simple unstable theory. ACFA has elimination of imaginaries. ACFA does not have quantifier elimination, but comes very close.

Completions and QE
A model $$(K,\sigma)$$ of ACFA is determined up to elementary equivalence by the isomorphism class of $$(K',\sigma')$$, where $$K'$$ is the algebraic closure of the prime field, and $$\sigma'$$ is the restriction of $$\sigma$$ to $$K$$. Conversely, if $$K'$$ is the algebraic closure of a prime field, and $$\sigma'$$ is any automorphism of $$K'$$, then some model of ACFA extends $$(K',\sigma')$$, so all the possibilities occur.

The upshot is that the completions of ACFA are in bijection with pairs $$(p,\sigma)$$, where $$p$$ is 0 or a prime, and where $$\sigma$$ is a conjugacy class in $$\operatorname{Gal}(\overline{K}/K)$$, where $$K$$ is the prime field of characteristic $$p$$.

ACFA does not have quantifier elimination. However, every formula is equivalent to one of the form $$\exists x : P(x,y) = 0 \wedge \phi(x;y),$$ where $$x$$ is a single variable, $$P(x,y)$$ is a polynomial, monic in $$x$$, and $$\phi(x;y)$$ is quantifier-free.

Relation to pseudofinite fields
If $$(K,\sigma) \models ACFA$$, the fixed field of $$\sigma$$ is pseudo-finite. More generally, it turns out that $$(K,\sigma^n) \models ACFA$$ for any nonzero integer $$n$$, so the fixed fields of the powers of $$\sigma$$ are also pseudofinite.

Elementary theory of the Frobenius
In the unpublished paper The Elementary Theory of the Frobenius, Hrushovski proves that ACFA is the limit theory of the Frobenius automorphism, in the same sense that the theory of pseudo-finite fields is the limit theory of finite fields.

More precisely, for each prime power $$q$$, let $$F_q$$ be the difference field whose underlying field is $$\overline{\mathbb{F}_q}$$, and whose automorphism is the $$q$$th power Frobenius map. Then Hrushovski proves that any non-principal ultraproduct of $$F_q$$'s models ACFA. This fact generalizes Ax's theorem that non-principal ultraproducts of finite fields are pseudo-finite. The part of Ax's proof that is hardest to generalize is the Lang-Weil estimates, so Hrushovski is forced to prove "twisted Lang-Weil" estimates, which have some independent interest with algebraic geometers.

Trichotomy in ACFA
Some analogue of the Zilber Trichotomy holds in ACFA (which is unstable). This was proven by Chatzidakis and Hrushovski, maybe, at least in the characteristic 0 case.

As an application, this can be used to give yet another proof of the Manin-Mumford conjecture.

ACFA and algebraic dynamics
Hrushovski's result on the elementary theory of the frobenius has been used to prove results in algebraic dynamics about periodic points, by Fakhruddin and someone else.

Also, the trichotomy in ACFA has been used by Medvedev and Scanlon to prove results in algebraic dynamics, about periodic varieties.