Bounded negativity of self-intersection numbers of Shimura curves in Shimura surfaces

Möller, Martin; Toledo, Domingo

doi:10.2140/ant.2015.9.897

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Bounded negativity of self-intersection numbers of Shimura curves in Shimura surfaces

Martin Möller and Domingo Toledo

Vol. 9 (2015), No. 4, 897–912

DOI: 10.2140/ant.2015.9.897

Abstract

Shimura curves on Shimura surfaces have been a candidate for counterexamples to the bounded negativity conjecture. We prove that they do not serve this purpose: there are only finitely many whose self-intersection number lies below a given bound.

Previously (Duke Math. J. 162:10 (2013), 1877–1894), this result was shown for compact Hilbert modular surfaces using the Bogomolov–Miyaoka–Yau inequality. Our approach uses equidistribution and works uniformly for all Shimura surfaces.

Keywords

bounded negativity, Shimura curves, self-intersections, equidistribution of Shimura curves

Mathematical Subject Classification 2010

Primary: 14G35

Secondary: 14J25, 22E40, 37A99, 53C55

Milestones

Received: 16 August 2014

Revised: 2 March 2015

Accepted: 7 April 2015

Published: 30 May 2015

Authors

Martin Möller
Institut für Mathematik Goethe-Universität Frankfurt Robert-Mayer-Strasse 6–8 D-60325 Frankfurt am Main Germany

Domingo Toledo
Department of Mathematics University of Utah 155 S 1400 E Salt Lake City, UT 84112 United States

Vol. 9, No. 4, 2015