%0 Journal Article
%T Transgressions of the Euler class and Eisenstein cohomology of GLN(Z)
%+ Département de Mathématiques et Applications - ENS Paris (DMA)
%+ Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586))
%+ Department of Mathematics (UCL London)
%A Bergeron, Nicolas
%A Charollois, Pierre
%A Garcia, Luis
%< avec comité de lecture
%@ 0289-2316
%J Japanese Journal of Mathematics
%I Springer Verlag
%V 15
%N 2
%P 311-379
%8 2020-09
%D 2020
%R 10.1007/s11537-019-1822-6
%Z Mathematics [math]/Number Theory [math.NT]
%Z Mathematics [math]/Geometric Topology [math.GT]Journal articles
%X 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 GL N (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 SL N (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 (GL N , GL 1). This suggests looking to reductive dual pairs (GL N , GL k) 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 don't deal with the most general cases and put a lot of emphasis on various examples that are often classical.
%G English
%2 https://ens.hal.science/hal-02886362/document
%2 https://ens.hal.science/hal-02886362/file/Takagi-final-JJM1822.pdf
%L hal-02886362
%U https://ens.hal.science/hal-02886362
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