Systoles of Arithmetic Hyperbolic Surfaces and 3-manifolds

TL;DR

This paper proves that for large systole bounds, almost all arithmetic hyperbolic 2- and 3-orbifolds with a fixed invariant trace field have systoles exceeding that bound, using number theory and algebraic geometry techniques.

Contribution

It establishes density one results for systole bounds in arithmetic hyperbolic orbifolds, extending previous work and providing new bounds related to area and geometric structures.

Findings

Density one of orbifolds with systole above a threshold

Stronger bounds when trace field is Q, depending on area

Density results for 3-orbifolds with small area geodesic surfaces

Abstract

Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds 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-orbifolds 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 $\mathbf{Q}$, using work of Granville and Soundararajan, we establish a stronger result that allows our constant lower bound $x_0$ 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 $2$-orbifolds.