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

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Arakelov Geometry and Diophantine Applications

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2276))

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Abstract

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.

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Acknowledgements

I am indebted to Huayi Chen, Emmanuel Peyre and Gaël Rémond for giving me the opportunity to participate in the summer school of the “Institut Fourier” in Grenoble, and their warm hospitality that made a kidney stone attack much less painful. Thanks as well to the students and other colleagues for attending the lectures and making encouraging comments on this circle of ideas.

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Authors and Affiliations

  1. CNRS—Institut de Mathématiques de Jussieu-Paris Rive Gauche, Paris, France

    Gerard Freixas i Montplet

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  1. Gerard Freixas i Montplet
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Correspondence to Gerard Freixas i Montplet .

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Editors and Affiliations

  1. Institut Fourier, Université Grenoble Alpes, Grenoble, France

    Emmanuel Peyre

  2. Institut Fourier, Université Grenoble Alpes, Grenoble, France

    Gaël Rémond

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Montplet, G.F.i. (2021). Chapter XI: The Arithmetic Riemann–Roch Theorem and the Jacquet–Langlands Correspondence. In: Peyre, E., Rémond, G. (eds) Arakelov Geometry and Diophantine Applications. Lecture Notes in Mathematics, vol 2276. Springer, Cham. https://doi.org/10.1007/978-3-030-57559-5_12

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