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 $4.01\log^2\Delta_{\mathbf K}$, $4\big(1+\big(2\pi e^{\gamma})^{-n_{\mathbf K}}\big)^2\log^2\Delta_{\mathbf K}$ 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.