Derived Langlands VII: The PSH Algebra of Products of General Linear Groups

TL;DR

This paper develops a PSH-like algebraic structure on groups related to products of general linear groups over finite fields, with potential extensions to p-adic local fields, aiming to understand properties of L-functions.

Contribution

It introduces a novel PSH-like structure on the R_+ groups of products of general linear groups, facilitating reductions that could help analyze L-functions.

Findings

Constructed a PSH-like algebraic framework for these groups.

Established reduction techniques to study properties of L-functions.

Potentially linked algebraic structures to functional equations of zeta and L-functions.

Abstract

In this article we put a very elaborate PSH-like structure on the $R_{+}(-)$ groups of products of finite general linear groups. This is not the case we want. Firstly one would really want the actual big PSH algebra of products of general linear groups with entries in a characteristic zero $p$-adic local field. There may be technical difficulties with this. However the $R_{+}(-)$ gadget for products of general linear groups with entries in a characteristic zero $p$-adic local field seems to work for us by allowing various reduction to compact open subgroups and reduction maps modulo different prime powers from there. These reductions may allow the verification of functional equations and analytic groups properties which characterise the Riemann zeta function and presumably similarly characterise the $2$-variable L-functions.