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

Abstract:

Let $\mathbb {{{K}}}$ 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 $\mathbb {{{K}}}$ and prove that the norm of the ideals in that set is $\leq (4-1/(2n))\log ^2\Delta$, except for a finite number of fields of degree $n\leq 4$. 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 $n\leq 4$, and to $(4-2/(3n)+3/(8n^2))\log ^2\Delta$ without exceptions.