Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

Cusp shapes of Hilbert–Blumenthal surfaces

Abstract: We introduce a new fundamental domain $$\mathscr {R}_n$$ for a cusp stabilizer of a Hilbert modular group $$\Gamma $$ over a real quadratic field $$K=\mathbb {Q}(\sqrt{n})$$. This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of $$\mathcal {H}^2\times \mathcal {H}^2$$. The region $$\mathscr {R}_n$$ is the product of $$\mathbb {R}^+$$ with a 3-dimensional tower $$\mathcal {T}_n$$ formed by deformations of lattices in the ring of integers $$\mathbb {Z}_K$$, and makes explicit the cusp cross section’s Sol 3-manifold structure and Anosov diffeomorphism. We include computer generated images and data illustrating various examples.

Coniveau over $p$-adic fields and points over finite fields

Abstract: If the $\ell$-adic cohomology of a projective smooth variety, defined over a $\frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $\ge 1$, then any model over the ring of integers of $K$ has a $k$-rational point. This slightly improves our earlier result math/0405318: we needed there the model to be regular (but then our result was more general: we obtained a congruence for the number of points, and $K$ could be local of characteristic $p>0$).

Dedekind's criterion for the monogenicity of a number field versus Uchida's and L\"uneburg's

Abstract: We compare three different characterizations, due respectively to R. Dedekind, K. Uchida, and H. Luneburg, of when $\mathbb Z[\theta]$ is the ring of integers of $\mathbb Q(\theta)$, and apply our results to some concrete $2$-towers of number fields.

On invariants related to non-unique factorizations in block monoids and rings of algebraic integers

Abstract: Let K be a number field, R its ring of integers and H the set of non-zero principal ideals of R. For each positive integer k the set Bk(H) C H denotes the set of principal ideals for which the associated block has at most k different factorizations. For the counting functions associated to these sets asymp­ totic formulae are known. These formulae involve constants that just depend on the ideal class group G of R. Starting from a known combinatorial description for these constants, we use tools from additive group theory, in particular the notion of Davenport's constant and a classical addition theorem, to investigate them. We determine their precise value in case G is an elementary group or a cyclic group of prime power order. For arbitrary G we derive (explicit) lower bounds .

Non-definability of rings of integers in most algebraic fields

Abstract: We show that the set of algebraic extensions $F$ of $\mathbb{Q}$ in which $\mathbb{Z}$ or the ring of integers $\mathcal{O}_F$ are definable is meager in the set of all algebraic extensions.