Chapter XI: The Arithmetic Riemann–Roch Theorem and the Jacquet–Langlands Correspondence

The arithmetic Riemann–Roch theorem refines both the algebraic geometric and differential geometric counterparts, and it is stated within the formalism of Arakelov geometry. For some simple Shimura varieties and automorphic vector bundles, the cohomological part of the formula can be understood via the theory of automorphic representations. Functoriality principles from this theory may then be applied to derive relations between arithmetic intersection numbers for different Shimura varieties. In this lectures we explain this philosophy in the case of modular curves and compact Shimura curves. This indicates that there is some relationship between the arithmetic Riemann–Roch theorem and trace type formulae.