Explicit bounds for generators of the class group

Loïc Grenié, Giuseppe Molteni

Published 2016-07-08Version 1

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 $12\mathcal L_{\mathbf K}^2$, and by $(4+o(1))\mathcal L_{\mathbf K}^2$ asymptotically, where $\mathcal L_{\mathbf K}$ is the logarithm of 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 a previous paper, we modified the algorithm, producing a new procedure which does even better. In this paper we prove explicit upper bounds for the size of the sets determined by both algorithms confirming that the first one has size $\asymp (\mathcal L_{\mathbf K}\log\mathcal L_{\mathbf K})^2$ and the second one $\asymp \mathcal L_{\mathbf K}^2$. Moreover, we show that $\mathcal C\ell_{\mathbf K}$ is generated by prime ideals whose norm is bounded by the minimum of $4.01\mathcal L_{\mathbf K}^2$, $4\big(1+\big(2\pi e^{\gamma})^{-n_{\mathbf K}}\big)^2\mathcal L_{\mathbf K}^2$ and $4\big(\mathcal L_{\mathbf K}+\log\mathcal L_{\mathbf K}-(\gamma+\log 2\pi)n_{\mathbf K}+1+(n_{\mathbf K}+1)\frac{\log(7\mathcal L_{\mathbf K})}{\mathcal L_{\mathbf K}}\big)^2$.