Explicit bounds for generators of the class group

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 $4.01{\log }^2{\Delta }_{\mathbf{K}}$, $4(1+(2\pi e^{\gamma }{)}^{-n_{\mathbf{K}}}{)}^2{\log }^2{\Delta }_{\mathbf{K}}$ and $4(\log {\Delta }_{\mathbf{K}}+\log \log {\Delta }_{\mathbf{K}}-(\gamma +\log 2\pi )n_{\mathbf{K}}+1+(n_{\mathbf{K}}+1)\frac{\log (7\log {\Delta }_{\mathbf{K}})}{\log {\Delta }_{\mathbf{K}}}{)}^2$. 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.

v5: corrected a couple of typos