Algebra & Number Theory

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

Martin Möller and Domingo Toledo

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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.

Article information

Algebra Number Theory, Volume 9, Number 4 (2015), 897-912.

Received: 16 August 2014
Revised: 2 March 2015
Accepted: 7 April 2015
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]
Secondary: 14J25: Special surfaces {For Hilbert modular surfaces, see 14G35} 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 37A99: None of the above, but in this section 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]

bounded negativity Shimura curves self-intersections equidistribution of Shimura curves


Möller, Martin; Toledo, Domingo. Bounded negativity of self-intersection numbers of Shimura curves in Shimura surfaces. Algebra Number Theory 9 (2015), no. 4, 897--912. doi:10.2140/ant.2015.9.897.

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