Abstract We show that every smooth projective curve over a finite field $k$ admits tame morphism to the line $k$. Furthermore, we construct with no such map when is an infinite perfect of characteristic two. Our work leads refinement Belyi theorem in positive characteristic, building on results Saïdi, Sugiyama–Yasuda, and Anbar–Tutdere.

Let X → S be an abelian scheme over a finite field. We show that for a set of rational primes l of density one, if q ≡ 1 mod l is sufficiently large, then the proportion of s ∈ S(Fq) for which l| ∣Xs(Fq) ∣∣ is at least 1/l−O(1/l2). We apply this to study class numbers of cyclic covers of the projective line.

We apply the semidefinite programming method to derive bounds for projective codes over a finite field.

— Let X be a smooth double cover of a geometrically ruled surface defined over a separably closed field of characteristic different from 2. The main result of this paper is a finite presentation of the 2-torsion in the Brauer group of X with generators given by central simple algebras over the function field of X and relations coming from the Néron-Severi group of X. The path to this result nat...

Let φ and ψ be endomorphisms of the projective line of degree at least 2, defined over a field F . From a dynamical perspective, a significant question is to determine whether φ and ψ are conjugate (or to answer the related question of whether a given rational function φ has a nontrivial automorphism). We construct efficient algorithms for computing the set of conjugating maps (resp., the group...

We show that the automorphism group of Drinfeld’s half-space over a finite field is the projective linear group of the underlying vector space. The proof of this result uses analytic geometry in the sense of Berkovich over the finite field equipped with the trivial valuation. We also take into account extensions of the base field.

We discuss representations of the projective line over a ring R with 1 in a projective space over some (not necessarily commutative) field K. Such a representation is based upon a (K,R)-bimodule U . The points of the projective line over R are represented by certain subspaces of the projective space P(K,U ×U) that are isomorphic to one of their complements. In particular, distant points go over...