Computing Singular Elements Modulo Squares

The group of singular elements was first introduced by Helmut Hasse and later it has been studied by numerous authors including such well known mathematicians as: Cassels, Furtwängler, Hecke, Knebusch, Takagi and of course Hasse himself; to name just a few. The aim of the present paper is to present algorithms that explicitly construct groups of singular and S-singular elements (modulo squares) in a global function field.

Wild sets in global function fields

Given a self-equivalence of a global function field, its wild set is the set of points where the self-equivalence fails to preserve parity of valuation. In this paper we describe structure of finite wild sets.

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]