A twisted theorem of Chebotarev

Abstract

We prove a function-field version of Chebotarev's density theorem in the framework of difference algebraic geometry by developing the notion of Galois coverings of generalized difference schemes, and using Hrushovski's twisted Lang-Weil estimate.We consider our approach to generalized difference algebra a major advance in its own right, and it should be of intrinsic interest in the difference algebra community. In view of the fact that the work of Wibmer [22] can be rewritten and possibly slightly sharpened using generalized difference algebra, as well as the elaborate study of generalized difference extensions along the lines of [20], we envisage applications in Galois theory of difference equations.Moreover, there are other naturally occurring contexts where it may be advantageous to study objects with several endomorphisms. In particular, as Tom Scanlon pointed out, a key step in most approaches to Manin-Mumford conjecture ([11], for example) is to find a difference polynomial equation that captures all the relevant torsion points, which is very hard to achieve with a single endomorphism. On the other hand, it is much easier to capture all of the torsion if one uses two endomorphisms.Apart from number theory and algebraic geometry, our results (especially Theorem 4.27) should be of interest in model-theory and logic since, in conjunction with the description of definable sets in terms of Galois stratifications from [20], they reduce the problem of counting points on definable sets over fields with powers of Frobenius to a calculation of (twisted) character sums. Thus, the present paper can be viewed as a conceptualization of ideas of [3,17] for fields with Frobenii. As already mentioned, Theorem 4.27 and its consequence Corollary 4.28 are used in our subsequent work on Galois stratifications in [20]. Generalized difference schemesIn this section, we wish to broaden the class of difference schemes in order to allow certain Galois actions. We shall henceforth refer to difference schemes with a single endomorphism as discussed in [12,13] as strict, or ordinary difference schemes, and we shall expand the term 'difference scheme' to include objects with multiple endomorphisms. Following a suggestion by the referee, we removed a number of highly technical verifications to make this section more readable. We direct the interested reader to the details [19].2.

Twisted Galois Stratification

“…where X i , i ∈ I is a partition of X into finitely many normal locally closed difference subschemes of X, each (Z i , Σ i ) → (X i , σ) is a Galois covering with some group (G i ,Σ) and C i is a conjugacy domain in Σ i , with all these notions precisely defined in [19]. The Galois formula associated with A is the realisation subfunctorà of X defined by the assignment…”

“…where (F, ϕ) is an algebraically closed difference field and the conjugacy class ϕ x ⊆ Σ is the local ϕ-substitution at x, as defined in [19,Section 4]. Our principal result in its algebraic-geometric incarnation is the following direct image theorem, stating that a direct image of a Galois formula by a morphism of finite transformal type is equivalent to a Galois formula over fields with Frobenii.…”

Direct twisted Galois stratification

“…We refer the reader interested in the comparison of direct Galois covers with Galois covers of difference schemes defined in [16] to Remark A.9.…”

“…Remark A.9. Suppose that (X, Σ)/(Y, σ) is a finite Galois cover of transformally integral difference schemes of finite transformal type over a transformal domain (R, ς) with group G as in [16] (the extension of associated function fields is algebraically finite). By σ-localising Y (and X), we can obtain a direct Galois cover with a rather special property that π1 () : G 1 → G 0 is an isomorphism (and both groups are isomorphic to G).…”

“…In order to afford the existence of Galois covers and to provide a notational device for discussing several possible lifts of the difference operator, we must make our framework flexible enough to include presentations of generalised difference schemes, developed in [16]. Having that in mind, the real difficulty of this project is to isolate a robust enough notion of a Galois cover which will perpetuate through a number of operations and constructions (cf.…”