Systoles of arithmetic hyperbolic surfaces and $3$-manifolds

Abstract

Our main result is that for any positive real number x0, the set of commensurability classes of arithmetic hyperbolic 2– or 3–manifolds with fixed invariant trace field k and systole bounded below by x0 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 Q, 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.