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

Nicolas Bergeron (1) , Pierre Charollois (2) , Luis Garcia (3)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
3 UCL London - Department of Mathematics

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.

Domaines: Théorie des nombres [math.NT]; Topologie géométrique [math.GT]

Nicolas Bergeron, Pierre Charollois, Luis Garcia. Transgressions of the Euler class and Eisenstein cohomology of GLN(Z). Japanese Journal of Mathematics, 2020, 15 (2), pp.311-379. ⟨10.1007/s11537-019-1822-6⟩. ⟨hal-02886362⟩