Local and global questions "beyond endoscopy"

TL;DR

This paper explores advanced ideas beyond the endoscopy program in automorphic forms, focusing on local and global questions, functoriality, and the connections between trace formulas and L-functions.

Contribution

It provides a broad survey of recent developments beyond endoscopy, highlighting the relationship between local-global aspects, functoriality, and the Braverman--Kazhdan program.

Findings

Connections between trace formulas and functoriality are being clarified.

The Braverman--Kazhdan program is gaining renewed interest.

Structural insights into comparisons of automorphic representations are emerging.

Abstract

The near-completion of the program of endoscopy poses the question of what lies next. This article takes a broad view of ideas beyond the program of endoscopy, highlighting the connections among them, and emphasizing the relationship between local and global aspects. Central among those ideas is the one proposed in a 2000 lecture of R. P. Langlands, aiming to extract from the stable trace formula of a group G the bulk of those automorphic representations in the image of the conjectural functorial lift corresponding to a morphism of L-groups ${^LH} \to {^LG}$. With the extension of the problem of functoriality to the "relative" setting of spherical varieties and related spaces, some structure behind such comparisons has started to reveal itself. In a seemingly unrelated direction, a program initiated by Braverman--Kazhdan, also around 2000, to generalize the Godement--Jacquet proof of the functional equation to arbitrary L-functions, has received renewed attention in recent years. We survey ideas and developments in this direction, as well, and discuss the relationship between the two programs.