Gerard Freixas (Paris, CNRS)
On the Riemann-Roch formula in Arakelov geometry and the Jacquet-Langlands
correspondence
We introduce a particular version of the Riemann-Roch formula in Arakelov
geometry, valid for sheaves of cusp forms on suitable modular curves,
equipped with their natural Petersson metrics. We show some consequences
of the formula. For instance, combined with the Jacquet-Langlands
correspondence, it leads to a comparison of arithmetic self-intersection
numbers studied by Kudla-Rapoport-Yang and Maillot-Rossler (compact
Shimura curve case) and Bost and Kuhn (modular curve case).