## Algebra & Number Theory

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

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.ant/1510842339

Digital Object Identifier
doi:10.2140/ant.2015.9.897

Mathematical Reviews number (MathSciNet)
MR3352823

Zentralblatt MATH identifier
1327.14122

#### Citation

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. https://projecteuclid.org/euclid.ant/1510842339

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