Abstract

These notes were written to be distributed to the audience of the first author Takagi lectures to be delivered June, 23 2018. These are based on a work-in-progress of the four authors.

In this work-in-progress we give a new construction of some Eisenstein classes for $\mathrm{GL}_N (\mathbf{Z})$ that were first considered by Nori [38] and Sczech [42]. The starting point of this construction is a theorem of Sullivan on the vanishing of the Euler class of $\mathrm{SL}_N (\mathbf{Z})$ 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 {\it regularized theta lift} for the reductive dual pair $(\mathrm{GL}_1 , \mathrm{GL}_N)$. This suggests looking to reductive dual pairs $(\mathrm{GL}_k , \mathrm{GL}_N)$ with $k \geq 1$ for possible generalizations of the Eisenstein cocycle. This leads to interesting arithmetic lifts.

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. Our primary hope is to show that our construction sheds some light on classical and new rationality questions in arithmetic.