Spectra of algebraic fields and subfields

An algebraic field extension of ℚ or ℤ/(p) may be regarded either as a structure in its own right, or as a subfield of its algebraic closure F (either ℚ or ℤ/(p)). We consider the Turing degree spectrum of F in both cases, as a structure and as a relation on F, and characterize the sets of Turing degrees that are realized as such spectra. The results show a connection between enumerability in the structure F and computability when F is seen as a subfield of F. © 2009 Springer Berlin Heidelberg

Weakly commensurable groups, with applications to differential geometry

The article contains a survey of our results on weakly commensurable arithmetic and general Zariski-dense subgroups, length-commensurable and isospectral locally symmetric spaces and of related problems in the theory of semi-simple agebraic groups. We have included a discussion of very recent results and conjectures on absolutely almost simple algebraic groups having the same maximal tori and finite-dimensional division algebras having the same maximal subfields.Comment: Improved exposition, updated bibliography. arXiv admin note: substantial text overlap with arXiv:1212.121

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