arXiv:1704.03853v3 [math.LO] 16 Nov 2017 TAME STRUCTURES VIA MULTIPLICATIVE CHARACTER SUMS ON VARIETIES OVER FINITE FIELDS MINH CHIEU TRAN Abstract. We study the model theory of (F; <χ) where the field F is an algebraic closure of a finite field and <χ is an ordering on the multiplicative group F × induced by a group embedding χ ∶ F × → C × . Using number-theoretic bounds on multiplicative character sums over finite fields and Weyl’s criterion for equidistribution, we establish a number of properties about the interaction between <χ and the underlying field structure. We axiomatize these properties using first-order logic, show that the resulting theory is model complete, and obtain an analogue of a theorem by Ax. 1. Introduction Pseudo-finite fields are important examples of tame structures in model theory; see [Cha97] for a survey. The study of these structures began with Ax, who used results about counting points on varieties over finite fields and Chebotarev’s density theo- rem to show that a field is pseudo-finite if and only if it is elementarily equivalent to a non-principal ultraproduct of finite fields. In this paper we show that related results about multiplicative character sums on varieties over finite fields yield tame structures in a rather different fashion. This answers a version of a question of van den Dries, Hrushovski and Kowalski which we loosely interpret as asking for applications of character and exponential sums in model theory. (However, we do not use results in [Kow07] as they suggested.) Throughout, F is an algebraic closure of a finite field and χ is a group embedding from F × to C × , where F × and C × are the multiplicative groups of F and the field of complex number C respectively. Let U (p) ⊆ C × be the group of roots of unity with order coprime to p when p is prime and the group of roots of unity when p is zero. Let T ⊆ C × be the unit circle. Then Imageχ = U (p) ⊆ T where p = char(F). We denote by < the natural ordering on the field of real numbers R. Identifying the interval [0, 1) ⊆ R with T via α ↦ e 2πiα , the above < induces cyclic orderings on T and U (p) for p either prime or zero which we also denote by <. Define < χ on F × to be the pullback of < on T by χ and view < χ as a binary relation on F. We will show that (F; < χ ) is model theoretically tame for all F and χ as above. We can think of the above (F; < χ ) as an amalgam of two simpler structures: the algebraically closed field F and the “cyclically ordered” group (F × ; < χ ). The latter can be identified via χ with (U (p) ; <) where p = char(F). This suggests studying the model theory of (F; < χ ) by first analyzing each of these two structures and then understanding the way they are “glued” together. Date : December 8, 2019. 2010 Mathematics Subject Classification. Primary 03C65; Secondary 03B25, 03C10, 03C64, 11T24, 12L12. 1