THE JACQUET–LANGLANDS CORRESPONDENCE AND THE ARITHMETIC RIEMANN–ROCH THEOREM FOR POINTED CURVES

Abstract

We show how the Jacquet–Langlands correspondence and the arithmetic Riemann–Roch theorem for pointed curves, relate the arithmetic self-intersection numbers of the sheaves of modular forms — with their Petersson norms — on modular and Shimura curves: these are equal modulo ∑l∈Sℚ log l, where S is a controlled set of primes. These quantities were previously considered by Bost and Kühn (modular curve case) and Kudla–Rapoport–Yang and Maillot–Roessler (Shimura curve case). By the work of Maillot and Roessler, our result settles a question raised by Soulé.