# Systoles of arithmetic hyberbolic surfaces and 3-manifolds

## Abstract

Our main result is that for any positive real number $x_0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-manifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-manifolds with invariant trace field $k$. 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 the rationals, using work of Granville and Soundararajan, we establish a stronger result that allows our constant lower bound to instead grow with the area/volume. 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-manifolds with a small area totally geodesic surface.

## Repository Citation

Linowitz, Benjamin, D.B. McReynolds, P. Pollack, and L. Thompson. 2017. “Systoles of arithmetic hyberbolic surfaces and 3-manifolds.” Mathematical Research Letters 24(5): 1497-1522.

## Publisher

International Press

## Publication Date

1-1-2017

## Publication Title

Mathematical Research Letters

## Department

Mathematics

## Document Type

Article

## DOI

https://dx.doi.org/10.4310/MRL.2017.v24.n5.a8

## Language

English

## Format

text