Galois criterion for torsion points of Drinfeld modules

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2, but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.

Two-cocycles and cleft extensions in left braided categories

We define two-cocycles and cleft extensions in categories that are not necessarily braided, but where specific objects braid from one direction, like for a Hopf algebra [Formula: see text] a Yetter–Drinfeld module braids from the left with [Formula: see text]-modules. We will generalize classical results to this context and give some application for the categories of Yetter–Drinfeld modules and [Formula: see text]-modules. In particular, we will describe liftings of coradically graded Hopf algebras in the category of Yetter–Drinfeld modules with these techniques.

Zsigmondy theorem for arithmetic dynamics induced by a drinfeld module

Let [Formula: see text] be a Drinfeld [Formula: see text]-module defined over a global function field [Formula: see text] Let [Formula: see text] be a non-torsion point of [Formula: see text] with infinite [Formula: see text]-orbit. For each [Formula: see text] write the ideal [Formula: see text] as a quotient of relatively prime integral ideals. We establish an analogue of the classical Zsigmondy theorem for the ideal sequence [Formula: see text] i.e. for all but finitely many [Formula: see text] there exists a prime ideal [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text]

Generalized Yetter–Drinfeld (quasi)modules and Yetter–Drinfeld–Long bi(quasi)modules for Hopf quasigroups

As generalizations of Yetter–Drinfeld module over a Hopf quasigroup, we introduce the notions of Yetter–Drinfeld–Long bimodule and generalize the Yetter–Drinfeld module over a Hopf quasigroup in this paper, and show that the category of Yetter–Drinfeld–Long bimodules [Formula: see text] over Hopf quasigroups is braided, which generalizes the results in Alonso Álvarez et al. [Projections and Yetter–Drinfeld modules over Hopf (co)quasigroups, J. Algebra 443 (2015) 153–199]. We also prove that the category of [Formula: see text] having all the categories of generalized Yetter–Drinfeld modules [Formula: see text], [Formula: see text] as components is a crossed [Formula: see text]-category.