Benjamin Linowitz

Modular Forms on Noncongruence Subgroups and Atkin–Swinnerton-Dyer Relations

We give new examples of noncongruence subgroups Γ ⊂ SL2(ℤ) whose space of weight-3 cusp forms S 3(Γ) admits a basis satisfying the Atkin–Swinnerton-Dyer congruence relations with respect to a weight-3 newform for a certain congruence subgroup.

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.

A Noncommutative Analogue of the Odlyzko Bounds and Bounds on Performance for Space-Time Lattice Codes

This paper considers space-time coding over several independently Rayleigh faded blocks. In particular, we will concentrate on giving upper bounds for the coding gain of lattice space-time codes as the number of blocks grow. This problem was previously considered in the single antenna case by Bayer-Fluckiger et al. in 2006. Crucial to their work was Odlyzkos bound on the discriminant of an algebraic number field, as this provides an upper bound for the normalized coding gain of number field codes. In the MIMO context natural codes are constructed from division algebras defined over number fields and the coding gain is measured by the discriminant of the corresponding (noncommutative) algebra. In this paper, we will develop analogues of the Odlyzko bounds in this context and show how these bounds limit the normalized coding gain of a very general family of division algebra based space-time codes. These bounds can also be used as benchmarks in practical code design and as tools to analyze asymptotic bounds of performance as the number of independently faded blocks increases.

On the isospectral orbifold–manifold problem for nonpositively curved locally symmetric spaces

An old problem asks whether a Riemannian manifold can be isospectral to a Riemannian orbifold with nontrivial singular set. In this short note we show that under the assumption of Schanuel’s conjecture in transcendental number theory, this is impossible whenever the orbifold and manifold in question are length-commensurable compact locally symmetric spaces of nonpositive curvature associated to simple Lie groups.

Locally Equivalent Correspondences@@@Correspondances Localement Équivalentes

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.

Local selectivity of orders in central simple algebras

Let B be a central simple algebra of degree n over a number field K, and L ⊂ B be a strictly maximal subfield. We say that the ring of integers 𝒪L is selective if there exists an isomorphism class of maximal orders in B no element of which contains 𝒪L. In the present work, we consider a local variant of the selectivity problem and applications. We first prove a theorem characterizing which maximal orders in a local central simple algebra contain the global ring of integers 𝒪L by leveraging the theory of affine buildings for SLr(D) where D is a local central division algebra. Then as an application, we use the local result and a local–global principle to show how to compute a set of representatives of the isomorphism classes of maximal orders in B, and distinguish those which are guaranteed to contain 𝒪L. Having such a set of representatives allows both algebraic and geometric applications. As an algebraic application, we recover a global selectivity result, and give examples which clarify the interesting role of partial ramification in the algebra.

Parametrizing Shimura subvarieties of {\mathrm{A}_1} Shimura varieties and related geometric problems

AbstractThis paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of

On Fields of Definition of Arithmetic Kleinian Reflection Groups II

{X_{a, b} = (\mathbf{H}^2)^a \times (\mathbf{H}^3)^b}