Computing singular elements modulo squares

TL;DR

This paper introduces algorithms for explicitly constructing groups of singular and S-singular elements modulo squares in global function fields, building on foundational work by Hasse and others.

Contribution

It provides new algorithms for constructing singular and S-singular element groups in global function fields, expanding computational tools in algebraic number theory.

Findings

Algorithms for constructing singular elements groups

Explicit methods for S-singular elements

Advances in computational algebra in function fields

Abstract

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, Furtw\"{a}ngler, 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.