Twisted Hilbert modular surfaces, arithmetic intersections and the Jacquet–Langlands correspondence

We study arithmetic intersections on 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.