Mathematical Research Letters

Volume 24 (2017)

Number 5

Systoles of arithmetic hyperbolic surfaces and $3$-manifolds

Pages: 1497 – 1522

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

Authors

Benjamin Linowitz (Department of Mathematics, Oberlin College, Oberlin, Ohio, U.S.A.)

D. B. McReynolds (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Paul Pollack (Department of Mathematics, University of Georgia, Athens, Ga., U.S.A.)

Lola Thompson (Department of Mathematics, Oberlin College, Oberlin, Ohio, U.S.A.)

Abstract

Our main result is that for all sufficiently large $x_0 \gt 0$, 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 a small area totally geodesic $2$-orbifolds.

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The authors would like to thank Ian Agol, Mikhail Belolipetsky, Bobby Grizzard, and Alan Reid for useful conversations on the material of this paper. The authors also thank the anonymous referees for comments that helped improve the mathematics and readability of this article. The first author was partially supported by NSF RTG grant DMS-1045119 and by an NSF Mathematical Sciences Postdoctoral Fellowship. The second author was partially supported by the NSF grants DMS-1105710 and DMS-1408458. The third author was partially supported by the NSF grant DMS-1402268. The fourth author was partially supported by a Max Planck Institute fellowship and by an AMS Simons Travel Grant.

Received 26 August 2015

Accepted 14 February 2017

Published 29 December 2017