22 research outputs found
Locally Equivalent Correspondences
Given a pair of number fields with isomorphic rings of adeles, we construct
bijections between objects associated to the pair. For instance we construct an
isomorphism of Brauer groups that commutes with restriction. We additionally
construct bijections between central simple algebras, maximal orders, various
Galois cohomology sets, and commensurability classes of arithmetic lattices in
simple, inner algebraic groups. We show that under certain conditions, lattices
corresponding to one another under our bijections have the same covolume and
pro-congruence completion. We also make effective a finiteness result of Prasad
and Rapinchuk.Comment: Final Version. To appear in Ann. Inst. Fourie
Systoles of Arithmetic Hyperbolic Surfaces and 3-manifolds
Our main result is that for all sufficiently large , the set of
commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed
invariant trace field and systole bounded below by has density one
within the set of all commensurability classes of arithmetic hyperbolic 2- or
3-orbifolds with invariant trace field . The proof relies upon bounds for
the absolute logarithmic Weil height of algebraic integers due to Silverman,
Brindza and Hajdu, as well as precise estimates for the number of rational
quaternion algebras not admitting embeddings of any quadratic field having
small discriminant. When the trace field is , using work of
Granville and Soundararajan, we establish a stronger result that allows our
constant lower bound to grow with the area. As an application, we
establish a systolic bound for arithmetic hyperbolic surfaces that is related
to prior work of Buser-Sarnak and Katz-Schaps-Vishne. Finally, we establish an
analogous density result for commensurability classes of arithmetic hyperbolic
3-orbifolds with small area totally geodesic -orbifolds.Comment: v4: 17 pages. Revised according to referee report. Final version. To
appear in Math. Res. Let
The Fourier coefficients of Eisenstein series newforms
In this article, we study the Fourier coefficients of Eisenstein series newforms. We obtain a sharp refinement of the strong multiplicity-one theorem by showing that the density of primes p for which the pth Hecke eigenvalues of two distinct Eisenstein series newforms differ is of the form 1/n for some n ≥ 2. Additionally, we show that if f is an Eisenstein series newform whose Fourier coefficients af (n) are real then there is a constant δ > 0 such that the sequence (af (n))n≤x has at least δx sign changes
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
Petits écarts entre idéaux premiers et spectres de longueurs de 3-variétés hyperboliques arithmétiques
In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, given any arithmetic hyperbolic 3-orbifold derived from a quaternion algebra, any finite subset of its geodesic length spectrum, and any , we produce infinitely many kS$, and have volumes lying in an interval of (universally) bounded length. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner