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

TL;DR

This paper introduces a new algorithm under the Generalized Riemann Hypothesis to compute class group generators with bounds on ideal norms that improve previous limits, especially for fields of degree up to 4.

Contribution

The paper presents a novel algorithm that achieves tighter bounds on the norms of ideals generating the class group, surpassing the 4 barrier for certain degrees under GRH.

Findings

Bound on ideal norms is improved to (4 - 1/(2n)) log^2 Δ for most fields.

For degree n ≤ 4, the bound slightly increases to (4 - 1/(2n) + 1/(2n^2)) log^2 Δ.

When class group order is odd, bounds are further improved to (4 - 2/(3n)) log^2 Δ and (4 - 2/(3n) + 3/(8n^2)) log^2 Δ.

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 $\leq (4-1/(2n))\log^2\Delta$, 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.