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

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.