Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups

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dc.creatorKatz, Mikhail G.
dc.creatorSchaps, Mary
dc.creatorVishne, Uzi
dc.date2005-05-01
dc.date2007-02-07
dc.date.accessioned2026-07-07T07:45:01Z
dc.date.available2026-07-07T07:45:01Z
dc.descriptionWe apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). The Fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field. We generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces. The key tool is a new trace estimate valid for an arbitrary ideal in a quaternion algebra. We obtain a particularly sharp bound for a principal congruence tower of Hurwitz surfaces (PCH), namely the 4/3-bound syspi_1(X_{\PCH}) > 4/3 \log(g(X_{\PCH})). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.
dc.description25 pages. To appear in Journal of Differential Geometry
dc.identifierhttps://arxiv.org/abs/math/0505007
dc.identifierhttp://arxiv.org/abs/math/0505007
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/123405
dc.subjectDifferential Geometry
dc.subjectNumber Theory
dc.subjectRings and Algebras
dc.subject53C23, 11R52, 16K20
dc.titleLogarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups
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