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Transgressions of the Euler class and Eisenstein cohomology of GLN(Z)

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Abstract

These notes were written to be distributed to the audience of the first author’s Takagi Lectures delivered June 23, 2018. These are based on a work-in-progress that is part of a collaborative project that also involves Akshay Venkatesh.

In this work-in-progress we give a new construction of some Eisenstein classes for GLN (Z) that were first considered by Nori [41] and Sczech [44]. The starting point of this construction is a theorem of Sullivan on the vanishing of the Euler class of SLN (Z) vector bundles and the explicit transgression of this Euler class by Bismut and Cheeger. Their proof indeed produces a universal form that can be thought of as a kernel for a regularized theta lift for the reductive dual pair (GLN, GL1). This suggests looking to reductive dual pairs (GLN, GLk) with k ≥ 1 for possible generalizations of the Eisenstein cocycle. This leads to fascinating lifts that relate the geometry/topology world of real arithmetic locally symmetric spaces to the arithmetic world of modular forms.

In these notes we do not deal with the most general cases and put a lot of emphasis on various examples that are often classical.

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Acknowledgements

N.B. would like to thank the Mathematical Society of Japan, the local organisers, and especially Professor Kobayashi, for their invitation and their kind hospitality in Kyoto. We all thank our collaborator Akshay Venkatesh as well as Javier Fresan for their comments and corrections on these notes. L.G. wishes to thank IHES for providing excellent conditions for research while this work was done. L.G. also acknowledges financial support from the ERC AAMOT Advanced Grant.

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Authors and Affiliations

  1. Département de Mathématiques et applications, École Normale Supérieure, Université PSL, F-75005, Paris, France

    Nicolas Bergeron

  2. Univ Paris Diderot, Sorbonne Université, Institut de Mathématiques de Jussieu—Paris Rive Gauche, CNRS, F-75005, Paris, France

    Pierre Charollois

  3. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

    Luis E. Garcia

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  1. Nicolas Bergeron
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  2. Pierre Charollois
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Correspondence to Nicolas Bergeron, Pierre Charollois or Luis E. Garcia.

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Communicated by: Toshiyuki Kobayashi

This article is based on the 21st Takagi Lectures that the first author delivered at Research Institute for Mathematical Sciences, Kyoto University on June 23, 2018.

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Bergeron, N., Charollois, P. & Garcia, L.E. Transgressions of the Euler class and Eisenstein cohomology of GLN(Z). Jpn. J. Math. 15, 311–379 (2020). https://doi.org/10.1007/s11537-019-1822-6

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  • DOI: https://doi.org/10.1007/s11537-019-1822-6

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