Exceptional covers and bijections on rational points.

*(English)*Zbl 1127.14023A finite finite generically étale morphism \(f \colon X \to Y\) of normal, geometrically irreducible varieties over a finite field \(\mathbb{F}_q\) is called exceptional cover if the diagonal is the only geometrically irreducible component of the fiber product \(X \times_YX\) which is defined over \(\mathbb{F}_q\). The paper contains the proof of the following result due to H.W. Lenstra: an exceptional morphism maps \(X(\mathbb{F}_q)\) bijectively onto \(Y(\mathbb{F}_q)\). If \(f\) is exceptional over \(\mathbb{F}_q,\) then it is exceptional over \(\mathbb{F}_{q^m}\) for infinitely many \(m\). It turns out that this property characterizes exceptional covers. Namely, the authors prove that if a finite separable morphism \(f \colon X \to Y\) between normal varieties over \(\mathbb{F}_q\) induces a surjective or injective map from \(X(\mathbb{F}_{q^m}) \to Y(\mathbb{F}_{q^m})\) for infinitely many \(m\), then \(f\) is exceptional.

In the case where \(X\) is a curve it suffices to test a single \(m\) larger than an explicit constant depending on \(q\), the degree \(n\) of \(f\), and the genus \(g_X\) of \(X\). Precisely, the authors prove that \(f\) is exceptional provided either \(f\) maps \(X(\mathbb{F}_q)\) injectively into \(Y(\mathbb{F}_q)\), and \(\sqrt{q} > 2n^2+4ng_X\), or \(f\) maps \(X(\mathbb{F}_q)\) surjectively onto \(Y(\mathbb{F}_q)\), and \(\sqrt{q} > n!(3g_X+3n)\).

The proofs are based on Galois theory, Chebotarev density theorem for covers of curves over finite field, and the Castelnuovo genus inequality.

At the end of the paper the authors give some examples of covers of curves \(X \to Y\) over \(\mathbb{F}_q\) which are injective but not surjective or surjective but not injective on rational points, and discuss the following conjecture: for a finite, geometrically étale map \(f \colon X \to Y\) of degree \(n \geq 2\) between two smooth projective varieties of dimension \(r\) over \(\mathbb{F}_q\), there exists a constant \(C\), depending only on \(n, r\), and the \(\ell\)-adic Betti numbers of \(X\), such that if \(q > C\) and \(f\) induces an injection or a surjection between \(X(\mathbb{F}_q)\) and \(Y(\mathbb{F}_q)\), then \(f\) is exceptional.

In the case where \(X\) is a curve it suffices to test a single \(m\) larger than an explicit constant depending on \(q\), the degree \(n\) of \(f\), and the genus \(g_X\) of \(X\). Precisely, the authors prove that \(f\) is exceptional provided either \(f\) maps \(X(\mathbb{F}_q)\) injectively into \(Y(\mathbb{F}_q)\), and \(\sqrt{q} > 2n^2+4ng_X\), or \(f\) maps \(X(\mathbb{F}_q)\) surjectively onto \(Y(\mathbb{F}_q)\), and \(\sqrt{q} > n!(3g_X+3n)\).

The proofs are based on Galois theory, Chebotarev density theorem for covers of curves over finite field, and the Castelnuovo genus inequality.

At the end of the paper the authors give some examples of covers of curves \(X \to Y\) over \(\mathbb{F}_q\) which are injective but not surjective or surjective but not injective on rational points, and discuss the following conjecture: for a finite, geometrically étale map \(f \colon X \to Y\) of degree \(n \geq 2\) between two smooth projective varieties of dimension \(r\) over \(\mathbb{F}_q\), there exists a constant \(C\), depending only on \(n, r\), and the \(\ell\)-adic Betti numbers of \(X\), such that if \(q > C\) and \(f\) induces an injection or a surjection between \(X(\mathbb{F}_q)\) and \(Y(\mathbb{F}_q)\), then \(f\) is exceptional.

Reviewer: Vasyl I. Andriychuk (Lviv)