  We study arithmetic intersections on twisted (quaternionic) Hilbert modular
surfaces and Shimura curves over a real quadratic field. Our first main result
is the determination of the degree of the top arithmetic Todd class of an
arithmetic twisted Hilbert modular surface. This quantity is then related to
the arithmetic volume of a Shimura curve, via the arithmetic
Grothendieck-Riemann-Roch theorem and the Jacquet-Langlands correspondence.
