Systoles of arithmetic hyperbolic surfaces and \(3\)-manifolds (Q1651756)

By work of Borel, the set of arithmetic hyperbolic structures on a closed orientable surface of genus at least two is a finite subset of the moduli space of hyperbolic metrics on the surface. The short geodesic conjecture asserts that there is a constant \(C > 0\), independent of the genus, such that the length of the shortest closed geodesic (the \textit{systole}) of an arithmetic hyperbolic surface is at least \(C\), and an analogous uniform lower bound is conjectured also for the systole of arithmetic hyperbolic 3-orbifolds. ``Our main result is that for all sufficiently large \(x_0 > 0\), the set of commensurability classes of arithmetic hyperbolic 2- and 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- and 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.'' ``Finally, we establish an analogous density result for commensurability classes of arithmetic hyperbolic 3-orbifolds with a small area totally geodesic 2-orbifold.''