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 smaller than $\log ^2\Delta _{\mathbf {K}}$ except for $7$ out of the $31292$ fields we tested.