June 23 (Sat), 2018 11:30--12:30, 15:20--16:20 Lecture Hall (Room No. 420) Research Institute for Mathematical Sciences Kyoto University, Kyoto, Japan |
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.