Complex Algebras of Arithmetic

An 'arithmetic circuit' is a labeled, acyclic directed graph specifying a sequence of arithmetic and logical operations to be performed on sets of natural numbers. Arithmetic circuits can also be viewed as the elements of the smallest subalgebra of the complex algebra of the semiring of natural numbers. In the present paper, we investigate the algebraic structure of complex algebras of natural numbers, and make some observations regarding the complexity of various theories of such algebras

Counting and effective rigidity in algebra and geometry

The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.Comment: v.2, 39 pages. To appear in Invent. Mat

Modular subvarieties of arithmetic quotients of bounded symmetric domains

Arithmetic quotients are quotients of bounded symmetric domains by arithmetic groups, and modular subvarieties of arithmetic quotients are themselves arithmetic quotients of lower dimension which live on arithmetic quotients, by an embedding induced from an inclusion of groups of hermitian type. We show the existence of such modular subvarieties, drawing on earlier work of the author. If $\Gamma$ is a fixed arithmetic subgroup, maximal in some sense, then we introduce the notion of ``$\Gamma$-integral symmetric'' subgroups, which in turn defines a notion of ``integral modular subvarieties'', and we show that there are finitely many such on an (isotropic, i.e, non-compact) arithmetic variety.Comment: 48 pages, also available at http://www.mathematik.uni-kl.de/~wwwagag/ LaTeX (e-mail: [email protected]