Breaking the 4 barrier for the bound of a generating set of the class group

Abstract

Let $K$ be a field of degree $n$ and discriminant with absolute value $\Delta$. Under the assumption of the validity of the Generalized Riemann Hypothesis, we provide a new algorithm to compute a set of generators of the class group of $K$ and prove that the norm of the ideals in that set is $\le (4-1/(2n)){\log }^2\Delta$, except for a finite number of fields of degree $n\le 4$. For those fields, the conclusion holds with the slightly larger limit $(4-1/(2n)+1/(2n^2)){\log }^2\Delta$. When the cardinality of $\mathcal{C}\ell$ is odd the bounds improve to $(4-2/(3n)){\log }^2\Delta$, again with finitely many exceptions in degree $n\le 4$, and to $(4-2/(3n)+3/(8n^2)){\log }^2\Delta$ without exceptions.

18 pages, 2 tables, 4 figures