The three smallest compact arithmetic hyperbolic 5-orbifolds

We determine the three hyperbolic 5-orbifolds of smallest volume among compact arithmetic orbifolds, and we identify their fundamental groups with hyperbolic Coxeter groups. This gives two different ways to compute the volume of these orbifolds.Comment: 11 page

Ray class fields of global function fields with many rational places

A general type of ray class fields of global function fields is investigated. The systematic computation of their genera leads to new examples of curves over finite fields with comparatively many rational points.Comment: Latex2e, 27 pages, 20 tables, revised version as submitted to Acta Arithmetic

Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's pp-rationality conjecture

In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements

On volumes of arithmetic quotients of PO(n,1), n odd

We determine the minimal volume of arithmetic hyperbolic orientable n-dimensional orbifolds (compact and non-compact) for every odd dimension n>3. Combined with the previously known results it solves the minimal volume problem for arithmetic hyperbolic n-orbifolds in all dimensions.Comment: 34 pages, final revision, to appear in Proc. LM

Computing ray class groups, conductors and discriminants

We use the algorithmic computation of exact sequences of Abelian groups to compute the complete structure of (ZK/m) ∗ for an ideal m of a number field K, as well as ray class groups of number fields, and conductors and discriminants of the corresponding Abelian extensions. As an application we give several number fields with discriminants less than previously known ones