Explicit bounds for generators of the class group

TL;DR

This paper establishes explicit bounds for generating the class group of a number field using prime ideals with norms bounded by functions of the discriminant, improving theoretical bounds and computational methods under GRH.

Contribution

It provides new explicit bounds for class group generators and introduces an efficient algorithm that outperforms previous bounds in practical computations.

Findings

Bounded prime ideal norms by explicit functions of discriminant.

Confirmed size of generating sets is approximately proportional to (\log ext{Δ} imes \log ext{log} ext{Δ})^2.

Proposed algorithm yields smaller generating sets in practice than previous methods.

Abstract

Assuming Generalized Riemann's Hypothesis, Bach proved that the class group $\mathcal C\!\ell_{\mathbf K}$ of a number field ${\mathbf K}$ may be generated using prime ideals whose norm is bounded by $12\log^2\Delta_{\mathbf K}$, and by $(4+o(1))\log^2\Delta_{\mathbf K}$ asymptotically, where $\Delta_{\mathbf K}$ is the absolute value of the discriminant of ${\mathbf K}$. Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates $\mathcal C\!\ell_{\mathbf K}$ and which performs better than Bach's bound in computations, but which is asymptotically worse. In this paper we show that $\mathcal C\!\ell_{\mathbf K}$ is generated by prime ideals whose norm is bounded by the minimum of $4.01\log^2\Delta_{\mathbf K}$, $4\big(1+\big(2\pi e^{\gamma})^{-n_{\mathbf K}}\big)^2\log^2\Delta_{\mathbf K}$ and $4\big(\log\Delta_{\mathbf K}+\log\log\Delta_{\mathbf K}-(\gamma+\log 2\pi)n_{\mathbf K}+1+(n_{\mathbf K}+1)\frac{\log(7\log\Delta_{\mathbf K})}{\log\Delta_{\mathbf K}}\big)^2$. Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedman's algorithms, confirming that it has size $\asymp (\log\Delta_{\mathbf K}\log\log\Delta_{\mathbf K})^2$. In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be experimentally smaller than $\log^2\Delta_{\mathbf K}$ except for 7 out of 31000 fields.