ABSTRACT
In characteristic p > 0, if a finite SU-rank formula is orthogonal to all formulas defining fixed fields, then the set it defines is also modular but is usually not stable. □ □ □
8.5. One can then show that the theory ACFA coincides with the set of sentences true in all but finitely many of the difference fields Fq. This is the analogue of Ax’s theorem, which states that the theory of pseudofinite fields coincides with the set of sentences true in almost all finite fields. Hrushovski’s proof gives more: it gives estimates on the size of finite definable sets. Before stating his result, let me remark that one can replace the scheme of axioms (3) of the theory ACFA by the apparently weaker scheme of axioms (3'). (3') If U and V are varieties o f dimension d defined over K, with V C U x U°
where
Thus the same holds in K. Note that K er(/) is infinite if and only if the size of the Ker(/^) is unbounded. H
Some of these families, however, do not originate directly from a simple algebraic group; their definition involves some automorphism of the field Fq. They are the so-called twisted finite simple groups. And they become definable in the structure (F l^g, oq). Using Theorem 8.4, one then also gets uniformity results for these families.
