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Computing Singular Elements Modulo Squares

Koprowski Przemysław

Fundamenta Informaticae

2021

Vol. 179, nr 3

227--338

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.

Linearly independent divisors on complex manifolds

Jelonek Z.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 1

25-30

Let X be a connected normal complex space and let D be a non-zero Cartier divisor on X with the support [...]. We show that if D is a principal divisor then the group H[sub 1](X \ [absolute value of D],Z) cannot be a torsion group. In particular the group H[sub1](H \ [absolute value of D],Z)] must be infinite. As a corollary we prove that simple Cartier divisors D[sub 1],...,D[sub r] on a complex manifold X are linearly independent in Cl(X), provided the group H[sub1](X \ [...] is a torsion group.