Vol. 9, No. 4, 2015

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ISSN: 1944-7833 (e-only)
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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
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